# The universal property of bordism
of commuting involutions

Markus Hausmann  and  Stefan Schwede<EMAIL_ADDRESS><EMAIL_ADDRESS>Mathematisches Institut, Universität Bonn, Germany

###### Abstract.

We propose a formalism to capture the structure of the equivariant bordism
rings of smooth manifolds with commuting involutions. We introduce the concept
of an oriented ${\operatorname{el}^{RO}_{2}}$-algebra, an algebraic structure
featuring representation graded rings for all elementary abelian 2-groups,
connected by restriction homomorphisms, a pre-Euler class, and an inverse Thom
class; this data is subject to one exactness property. Besides equivariant
bordism, oriented global ring spectra also give rise to oriented
${\operatorname{el}^{RO}_{2}}$-algebras, so examples abound. Inverting the
inverse Thom classes yields a global 2-torsion group law. In this sense, our
oriented ${\operatorname{el}^{RO}_{2}}$-algebras are delocalized
generalizations of global 2-torsion group laws.

Our main result shows that equivariant bordism for elementary abelian 2-groups
is an initial oriented ${\operatorname{el}^{RO}_{2}}$-algebra. Several other
interesting equivariant homology theories can also be characterized, on
elementary abelian 2-groups, by similar universal properties. We prove that
stable equivariant bordism is an initial
${\operatorname{el}^{RO}_{2}}$-algebra with an invertible orientation; that
Bredon homology with constant mod 2 coefficients is an initial
${\operatorname{el}^{RO}_{2}}$-algebra with an additive orientation; and that
Borel equivariant homology with mod 2 coefficients is an initial
${\operatorname{el}^{RO}_{2}}$-algebra with an orientation that is both
additive and invertible.
2020 MSC: 55N22, 55N91, 55P91, 57R85

###### Contents

  1. 1 Oriented ${\operatorname{el}^{RO}_{2}}$-algebras
  2. 2 Oriented global ring spectra
  3. 3 The universal property of Bredon homology
  4. 4 The associated 2-torsion formal group law
  5. 5 Geometric fixed points
  6. 6 Equivariant bordism
  7. 7 The universal property of equivariant bordism
  8. 8 Invertibly oriented ${\operatorname{el}^{RO}_{2}}$-algebras and global 2-torsion group laws
  9. 9 Epilogue: reproving classical results on equivariant bordism

## Introduction

The purpose of this paper is to unveil the universal property of the
equivariant bordism rings for elementary abelian 2-groups. More precisely, we
will show that equivariant bordism of manifolds with commuting involutions is
an initial example of a specific kind of algebraic structure that we call an
oriented ${\operatorname{el}^{RO}_{2}}$-algebra. The superscript ‘$RO$’ refers
to the feature that gradings by real representations are hard wired into our
theory. Our formalism also allows to characterize several other interesting
equivariant homology theories, on elementary abelian 2-groups, by similar
universal properties in the category of oriented
${\operatorname{el}^{RO}_{2}}$-algebras; examples include stable equivariant
bordism, Bredon homology with constant mod 2 coefficients, and Borel
cohomology with mod 2 coefficients. In short, stable equivariant bordism is
initial among oriented ${\operatorname{el}^{RO}_{2}}$-algebras equipped with
an invertible orientation; Bredon homology is initial among oriented
${\operatorname{el}^{RO}_{2}}$-algebras equipped with an additive orientation;
and Borel cohomology is initial among oriented
${\operatorname{el}^{RO}_{2}}$-algebras equipped with an orientation that
simultaneously invertible and additive.

The key new insight that we wish to promote in this paper is that the
algebraic structure of an orientable ${\operatorname{el}^{RO}_{2}}$-algebra
arises very naturally in the study of equivariant homology theories for
elementary abelian 2-groups, and that it greatly clarifies and conceptualizes
the ‘true’ nature of several prominent equivariant theories. Indeed, for all
the theories mentioned in the previous paragraph, there is a vast literature
of explicit calculations, in particular for the group with two elements.
Examples include [2, 4, 5, 6, 7, 10, 12, 15, 28, 29, 30] for equivariant
bordism, [9, 15, 28] for stable equivariant bordism, and [3, 11, 14, 18, 19,
20, 21, 22, 26, 27] for Bredon homology and cohomology. Many of the answers
are in terms of lists of generators and relations. While often quite explicit,
the present authors do not find many of these calculations conceptually
enlightening, and contemplating the answers left us wondering what it actually
means to ‘calculate’ or understand such kinds of mathematical structures.
Furthermore, none of the previous results we are aware of relate to Quillen’s
algebraic interpretation [24] of the bordism ring $\mathcal{N}_{*}$ as the
$2$-torsion Lazard ring, carrying the universal formal group law with trivial
$2$-series.

Our approach shifts the emphasis away from explicit calculations of
equivariant bordism or cohomology rings for individual groups of equivariance;
we focus on the question: ‘What do the equivariant rings represent?’ A lesson
we learned is that for getting interesting answers, one might want to:

  * •

consider all elementary abelian 2-groups simultaneously, keeping track of the
contravariant functoriality in group homomorphisms, and

  * •

remember and exploit the $RO$-grading by real representations.

We implement these two aspects in the structure an oriented
${\operatorname{el}^{RO}_{2}}$-algebra. Central to this notion are two
distinguished classes of elements, the ‘pre-Euler classes’ $a_{\lambda}$ and
the ‘inverse Thom classes’ $t_{\lambda}$, indexed by all nontrivial characters
$\lambda$ of elementary abelian $2$-groups. The structure is subject to one
key regularity property: multiplication by the pre-Euler classes of nontrivial
characters must be injective, with cokernel given by the value at the kernel
of the given character, while the classes $t_{\lambda}$ must restrict to the
unit at the trivial group. In the case of equivariant bordism, $a_{\lambda}$
equals the bordism class of the inclusion $\\{0\\}\longrightarrow S^{\lambda}$
of the origin into the representation sphere associated to $\lambda$, while
$t_{\lambda}$ is given by the bordism class of the identity map
$S^{\lambda}\longrightarrow S^{\lambda}$. In the case of Bredon homology, the
classes $a_{\lambda}$ and $t_{\lambda}$ generate the entire representation-
graded ring, subject to certain explicit polynomial relations [18]. The
precise definition of an oriented ${\operatorname{el}^{RO}_{2}}$-algebra will
be given in Definition 1.4. Notably, the category of oriented
${\operatorname{el}^{RO}_{2}}$-algebras contains the category of $2$-torsion
formal group laws as a reflective subcategory via the ‘complete
$t$-invertible’ objects, see Section 8. Hence, in very loose terms, an
oriented ${\operatorname{el}^{RO}_{2}}$-algebra can be viewed as a delocalized
and decompleted version of a $2$-torsion formal group law.

After this attempt to provide motivation for our formalism, we now list our
main results. The first of these, to be proved as Theorem 7.14 below, makes
the title of this paper precise:

Theorem A. Equivariant bordism is an initial oriented
${\operatorname{el}^{RO}_{2}}$-algebra.

We emphasize that in contrast to all previous results that we are aware of,
Theorem A characterizes geometric equivariant bordism in absolute terms rather
than relative to the non-equivariant bordism ring $\mathcal{N}_{*}$. In
particular, no a priori connection to formal group laws, nor any
$\mathcal{N}_{*}$-algebra structure, are directly built into the definition of
an oriented ${\operatorname{el}^{RO}_{2}}$-algebra. Still, the regularity
properties of the pre-Euler classes together with the functoriality in
elementary abelian $2$-groups indirectly induce nontrivial structure also over
the trivial group, namely that of a $2$-torsion formal group law. As such,
Theorem A can be viewed as a refinement of Quillen’s theorem [24]; however,
our Theorem A does not provide an independent proof of Quillen’s result, as we
rely on it as central input.

We now turn to our next result. Stable equivariant bordism is a certain
localization of equivariant bordism first considered by Bröcker and Hook [9,
§2]. We call an orientation of an ${\operatorname{el}^{RO}_{2}}$-algebra
invertible if the inverse Thom classes for all characters of elementary
abelian 2-groups act invertibly, see Definition 8.1. Stable equivariant
bordism is, essentially by design, the oriented
${\operatorname{el}^{RO}_{2}}$-algebra obtained from equivariant bordism by
inverting all those inverse Thom classes. The following result is thus an
immediate corollary of Theorem A:

Theorem B. Stable equivariant bordism is an initial invertibly oriented
${\operatorname{el}^{RO}_{2}}$-algebra.

Theorem B will be proved as Corollary 8.6 below. As we explain at the end of
Section 8, combining Theorem B with the main result of Bröcker and Hook [9]
yields an independent proof of a recent result of the first author, namely
that the global 2-torsion group law carried by the global Thom spectrum
$\mathbf{MO}$ is initial, see [17, Theorem D]. In fact, this earlier result of
the first author was a key motivation for the present project, and some of our
arguments are inspired by arguments from [17].

The next interesting example is Bredon homology with coefficients in the
constant Mackey functor with value ${\mathbb{F}}_{2}$; by definition, the
${\mathbb{Z}}$-graded part of this theory is a copy of ${\mathbb{F}}_{2}$
concentrated in dimension 0. These Bredon homology groups have attracted a
certain amount of attention, and we listed specific references with many
explicit calculations above. In this paper we show an algebraic universal
property for the coefficients of Bredon homology. The oriented
${\operatorname{el}^{RO}_{2}}$-algebra of Bredon homology has a unique
orientation, and this orientation is additive in the sense of Definition 3.3.
We use the adjective ‘additive’ because the formal group law arising from an
additive orientation is the additive one, see Example 4.8.

Theorem C. Bredon homology with constant ${\mathbb{F}}_{2}$-coefficients is an
initial additively oriented ${\operatorname{el}^{RO}_{2}}$-algebra.

Theorem C is proved as Theorem 3.5 below. In our paper [18], we provide an
explicit presentation of the representation-graded Bredon homology rings with
constant ${\mathbb{F}}_{2}$-coefficients.

The last example whose universal property we discuss is the mod 2 global Borel
spectrum $b(H{\mathbb{F}}_{2})$, representing Borel equivariant cohomology
with ${\mathbb{F}}_{2}$-coefficients. The associated
${\operatorname{el}^{RO}_{2}}$-algebra has a unique orientation, and this
unique orientation is both additive and invertible. We prove in Theorem 8.8
that mod 2 global Borel cohomology is characterized by such an orientation:

Theorem D. Borel cohomology with ${\mathbb{F}}_{2}$-coefficients is initial
among oriented ${\operatorname{el}^{RO}_{2}}$-algebras whose orientation is
both additive and invertible.

In Section 9 we explain how our theory can be used to reprove various previous
results on equivariant bordism, in particular the Conner–Floyd exact sequence
[12], Alexander’s description [2] of an explicit basis over $\mathcal{N}_{*}$
for the bordism ring of involutions, and Firsching’s pullback square [15]. In
all cases, the results in fact hold for a large class of oriented
${\operatorname{el}^{RO}_{2}}$-algebras, and not just for the initial one
given by equivariant bordism.

Acknowledgments. The authors are members of the Hausdorff Center for
Mathematics at the University of Bonn (DFG GZ 2047/1, project ID 390685813). A
substantial part of the work for this paper was done while the second author
spent the summer term 2023 on sabbatical at Stockholm University, with
financial support from the Knut and Alice Wallenberg Foundation; the second
author would like to thank SU for the hospitality and stimulating atmosphere
during this visit.

##  1\. Oriented ${\operatorname{el}^{RO}_{2}}$-algebras

In this section we introduce our main characters, the oriented
${\operatorname{el}^{RO}_{2}}$-algebras. One of the key points of this paper
is the insight that to unveil the universal property of equivariant bordism
for elementary abelian 2-groups, one needs to take the grading by
representations into account. More precisely, the proper book keeping needs
the ‘effective’ part of the $RO$-grading, i.e., the part that keeps track of
the integer graded bordism groups of representation spheres.

We start by fixing notation and terminology to deal with the $RO$-grading. For
every finite group $G$, we define a submonoid $I_{G}$ of the real
representation ring $RO(G)$ by

$I_{G}\ =\ \\{m\in RO(G)\ :\ m=k-V\text{ for some $k\in{\mathbb{Z}}$ and some
$G$-representation $V$}\\}\ .$

So the elements of $I_{G}$ are formal differences of integers and actual
representations. These submonoids are closed under restriction along
homomorphisms $\alpha:K\longrightarrow G$ between finite groups, i.e.,
$\alpha^{*}:RO(G)\longrightarrow RO(K)$ sends $I_{G}$ to $I_{K}$. We emphasize
that unless the group $G$ is trivial, the abelian monoid $I_{G}$ is not a
group.

######  Construction 1.1 (The category ${\operatorname{el}^{RO}_{2}}$).

We introduce a category ${\operatorname{el}^{RO}_{2}}$ of effective
representations of elementary abelian 2-groups. Objects of
${\operatorname{el}^{RO}_{2}}$ are pairs $(A,m)$, where $A$ is an elementary
abelian 2-group and $m\in I_{A}$. Morphisms from $(B,n)$ to $(A,m)$ in the
category ${\operatorname{el}^{RO}_{2}}$ are group homomorphisms
$\alpha:B\longrightarrow A$ such that $\alpha^{*}(m)=n$.

We give the category ${\operatorname{el}^{RO}_{2}}$ a symmetric monoidal
structure as follows. On objects, we set

$(A,m)\times(B,n)\ =\ (A\times B,m\oplus n)\ ,$

where

$m\oplus n\ =\ p_{1}^{*}(m)+p_{2}^{*}(n)$

is the external sum, with $p_{1}:A\times B\longrightarrow A$ and
$p_{2}:A\times B\longrightarrow B$ the projections to the two factors. On
morphisms, the symmetric monoidal structure is given by product of
homomorphisms. The symmetry and associativity isomorphisms are the symmetry
and associativity isomorphism for the product of groups. The unit object is
$(1,0)$, the pair consisting of the trivial group $1$ and its zero
representation. A symmetric monoidal structure on a category is at the same
time a symmetric monoidal structure on its opposite.

######  Definition 1.2.

An ${\operatorname{el}^{RO}_{2}}$-algebra is a lax symmetric monoidal functor

$X\ :\ ({\operatorname{el}^{RO}_{2}})^{\operatorname{op}}\ \longrightarrow\
{\mathbb{F}}_{2}\text{-mod}$

to the category of ${\mathbb{F}}_{2}$-vector spaces under tensor product. A
morphism of ${\operatorname{el}^{RO}_{2}}$-algebras is a monoidal
transformation.

Since symmetry is part of the definition of an
${\operatorname{el}^{RO}_{2}}$-algebra, a more precise name for these objects
would have been commutative ${\operatorname{el}^{RO}_{2}}$-algebras. Since we
will not consider non-commutative ${\operatorname{el}^{RO}_{2}}$-algebras
(i.e., non-symmetric lax monoidal functors), we have opted for the simpler
name.

Expanding the definition reveals that an
${\operatorname{el}^{RO}_{2}}$-algebra $X$ consists of

  * •

${\mathbb{F}}_{2}$-vector spaces $X(A,m)$ for every elementary abelian 2-group
$A$ and every $m\in I_{A}$,

  * •

a restriction homomorphism

$\alpha^{*}\ :\ X(A,m)\ \longrightarrow\ X(B,\alpha^{*}(m))\ ,$

for every group homomorphism $\alpha:B\longrightarrow A$ and every $m\in
I_{A}$,

  * •

associative and commutative pairings

$\times\ :\ X(A,m)\otimes X(B,n)\ \longrightarrow\ X(A\times B,m\oplus n)$

with a multiplicative unit $1\in X(1,0)$.

Moreover, the restriction homomorphisms must be contravariantly functorial and
strictly compatible with the multiplicative structure. A morphism
$f:X\longrightarrow Y$ of ${\operatorname{el}^{RO}_{2}}$-algebras consists of
additive maps $f(A,m):X(A,m)\longrightarrow Y(A,m)$ such that $f(1,0)(1)=1$
and

$f(A\times B,m\oplus n)(x\times y)\ =\ f(A,m)(x)\times f(B,n)(y)$

for all $x\in X(A,m)$ and $y\in X(B,n)$.

The multiplicative structure could alternatively be given in ‘internal form’

$\cdot\ :\ X(A,m)\otimes X(A,n)\ \longrightarrow\ X(A,m+n)\ ,$

as the composite

$X(A,m)\otimes X(A,n)\ \xrightarrow{\ \times\ }\ X(A\times A,m\oplus n)\
\xrightarrow{\Delta^{*}}X(A,m+n)\ ,$

where $\Delta:A\longrightarrow A\times A$ is the diagonal. For fixed $A$ and
varying $m$, this makes the collection of groups $X(A,m)$ into a commutative
$I_{A}$-graded ${\mathbb{F}}_{2}$-algebra. We will often employ the notation
$X(A,\star)$ for this $I_{A}$-graded ${\mathbb{F}}_{2}$-algebra. The abelian
groups $X(A,k)$ for $k\in{\mathbb{Z}}$ form a commutative
${\mathbb{Z}}$-graded ${\mathbb{F}}_{2}$-algebra, the integer graded subring
of this $I_{A}$-graded ${\mathbb{F}}_{2}$-algebra $X(A,\star)$, which we
denote by $X(A)_{*}$. The external multiplication can be recovered from the
internal multiplication as the composite

$X(A,m)\otimes X(B,n)\ \xrightarrow{p_{1}^{*}\otimes p_{2}^{*}}\ X(A\times
B,p_{1}^{*}(m))\otimes X(A\times B,p_{2}^{*}(n))\ \xrightarrow{\ \cdot\
}X(A\times B,m\oplus n)\ .$

######  Remark 1.3.

The cautious reader might wonder why we consider functors on isomorphism
classes of representations, and why we abuse notation and use the same symbol
for a representation and for its class in $RO(G)$. In other words: why does
our theory not take automorphisms of representations into account? The reason
is that the automorphisms of real representations, and even all equivariant
homotopy self-equivalences of linear spheres, are invisible to the theories we
consider. We explain this in detail in Theorem 2.3, but here is already a
brief sketch. All theories relevant to us arise from global ring spectra $E$
in which $\operatorname{tr}_{1}^{C}(1)=0$, i.e., the transfer from the trivial
group to the two element group $C$ vanishes in $\pi_{0}^{C}(E)$. This property
has strong consequences for the coefficients of the theory: for every compact
Lie group $G$, the unit ring map
$\pi_{0}^{G}({\mathbb{S}})\longrightarrow\pi_{0}^{G}(E)$ from the
$G$-equivariant 0-stem sends all units to 1. Consequently, all $G$-equivariant
homotopy self-equivalences of spheres of real $G$-representations induce the
identity in the $G$-homology theory represented by $E$.

In the following we write $C=\\{\pm 1\\}$ for the group with two elements. A
character of a group $G$ is a homomorphism $\lambda:G\longrightarrow C$. We
shall consistently confuse such characters with the associated 1-dimensional
$G$-representation on ${\mathbb{R}}$ with $G$-action $g\cdot x=\lambda(g)\cdot
x$, and with the class of this representation in the representation ring
$RO(G)$. We write $\sigma$ for the sign representation of $C$ on
${\mathbb{R}}$, corresponding to the identity character of $C$.

######  Definition 1.4 (Orientable ${\operatorname{el}^{RO}_{2}}$-algebras).

An orientable ${\operatorname{el}^{RO}_{2}}$-algebra is a pair $(X,a)$
consisting of an ${\operatorname{el}^{RO}_{2}}$-algebra $X$ and a class $a\in
X(C,-\sigma)$ with the following property: for every elementary abelian
2-group $A$, every $m\in I_{A}$ and every nontrivial $A$-character $\lambda$,
the sequence

$0\ \longrightarrow\ X(A,m)\ \xrightarrow{a_{\lambda}\cdot-}\ X(A,m-\lambda)\
\xrightarrow{\operatorname{res}^{A}_{K}}\
X(K,\operatorname{res}^{A}_{K}(m)-1)\ \longrightarrow\ 0$

is exact, where $K$ is the kernel of $\lambda$, and
$a_{\lambda}=\lambda^{*}(a)$. We refer to the class $a$ as the pre-Euler class
of the orientable ${\operatorname{el}^{RO}_{2}}$-algebra. A morphism of
orientable ${\operatorname{el}^{RO}_{2}}$-algebras is a morphism of
${\operatorname{el}^{RO}_{2}}$-algebras that takes the pre-Euler class to the
pre-Euler class.

An easy but useful fact is that a morphism of orientable
${\operatorname{el}^{RO}_{2}}$-algebras is already an isomorphism if its
restriction to all the integer graded subrings is:

######  Proposition 1.5.

Let $f:X\longrightarrow Y$ be a morphism of orientable
${\operatorname{el}^{RO}_{2}}$-algebras. If all the homomorphisms of
${\mathbb{Z}}$-graded rings $f(A)_{*}:X(A)_{*}\longrightarrow Y(A)_{*}$ are
bijective, then $f$ is an isomorphism of
${\operatorname{el}^{RO}_{2}}$-algebras.

###### Proof.

We show that for all elementary abelian 2-groups $A$, all $k\in{\mathbb{Z}}$
and all $A$-representations $W$ with $W^{A}=0$, the map
$f(A,k-W):X(A,k-W)\longrightarrow Y(A,k-W)$ is an isomorphism. We argue by
induction on the dimension of $W$. For $W=0$ the claim is true by hypothesis.
If $W$ is nonzero, we choose a nontrivial $A$-character $\lambda$ such that
$W=V\oplus\lambda$. Then the outer two maps in the following commutative
diagram are isomorphisms by induction, because $V$ has smaller dimension than
$W$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X(A,k-V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f(A,k-V)}$$\scriptstyle{a_{\lambda}\cdot-}$$\textstyle{X(A,k-W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f(A,k-W)}$$\scriptstyle{\operatorname{res}^{A}_{K}}$$\textstyle{X(K,k-1-V|_{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f(K,k-1-V|_{K})}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y(A,k-V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{\lambda}\cdot-}$$\textstyle{Y(A,k-W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{A}_{K}}$$\textstyle{Y(K,k-1-V|_{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Since both rows are exact, the middle map is an isomorphism. ∎

For an orientable ${\operatorname{el}^{RO}_{2}}$-algebra $(X,a)$ we already
employed the notation $a_{\lambda}=\lambda^{*}(a)$ for $A$-characters
$\lambda$. Then $a_{\lambda}=0$ when $\lambda$ is the trivial character; and
multiplication by $a_{\lambda}$ is injective for all nontrivial characters
$\lambda$. If $V$ is an $A$-representation, we define

$a_{V}\ =\ \prod_{\lambda\in\operatorname{Hom}(A,C)}a_{\lambda}^{m_{\lambda}}\
\in\ X(A,-V)\ ,$

where $m_{\lambda}$ is the multiplicity of the character $\lambda$ in $V$.
Then $a_{V}=0$ whenever $V$ has nontrivial $A$-fixed points; and
multiplication by $a_{V}$ is injective whenever $V^{A}=0$. These $a$-classes
are multiplicative and natural for restriction homomorphisms, i.e.,

$a_{V}\cdot a_{W}\ =\ a_{V\oplus W}\text{\quad and\quad}\beta^{*}(a_{V})\ =\
a_{\beta^{*}(V)}$

for all homomorphisms $\beta:B\longrightarrow A$ between elementary abelian
2-groups.

In an orientable ${\operatorname{el}^{RO}_{2}}$-algebra, divisibility by the
$a$-classes of general representations can be tested separately on the
isotypical summands. The next proposition makes this precise.

######  Proposition 1.6.

Let $(X,a)$ be an orientable ${\operatorname{el}^{RO}_{2}}$-algebra, and let
$A$ be an elementary abelian 2-group.

  1. (i)

Let $\lambda$ be a nontrivial $A$-character with kernel $K$. Let $x$ and $y$
be homogeneous elements of $X(A,\star)$. Suppose that
$\operatorname{res}^{A}_{K}(y)$ is a non zero-divisor in $X(K,\star)$. Then
$x$ is divisible by $a_{\lambda}^{n}$ if and only if $xy$ is divisible by
$a_{\lambda}^{n}$.

  2. (ii)

Let $V$ be an $A$-representation with trivial $A$-fixed points. Let $z$ be an
homogeneous element of $X(A,\star)$ that is divisible by
$a_{\lambda}^{m_{\lambda}}$ for every nontrivial character $\lambda$ of $A$,
where $m_{\lambda}$ is the multiplicity of $\lambda$ in $V$. Then $z$ is
divisible by $a_{V}$.

###### Proof.

(i) If $x$ is divisible by $a_{\lambda}^{n}$, then clearly so is $xy$. We
prove the other direction by induction on $n$. The statement is vacuous for
$n=0$. Now let $n\geq 1$. By the induction hypothesis we know that $x$ is
divisible by $a^{n-1}_{\lambda}$. Hence there exists an element $x^{\prime}$
such that $x=x^{\prime}a^{n-1}_{\lambda}$. Since
$x^{\prime}ya^{n-1}_{\lambda}=xy$, and $x^{\prime}y$ is the unique element
with this property, we know that $x^{\prime}y$ must be divisible by
$a_{\lambda}$. Hence the restriction
$\operatorname{res}^{A}_{K}(x^{\prime}y)=\operatorname{res}^{A}_{K}(x^{\prime})\operatorname{res}^{A}_{K}(y)$
is trivial. Since $\operatorname{res}^{A}_{K}(y)$ is a non zero-divisor, this
implies that $\operatorname{res}^{A}_{K}(x^{\prime})=0$. Hence $x^{\prime}$ is
divisible by $a_{\lambda}$, and therefore $x$ is divisible by
$a_{\lambda}^{n}$. This finishes the proof.

(ii) We argue by induction on the number of nontrivial isotypical summands of
$V$. There is nothing to show if $V=0$. Otherwise we let $\chi$ be a
nontrivial $A$-character that occurs in $V$. Then $V=\chi^{m_{\chi}}\oplus U$
for an $A$-representation $U$ with fewer nontrivial isotypical summands. We
let $z\in X(A,m)$ be divisible by $a_{\lambda}^{m_{\lambda}}$ for all
nontrivial $A$-characters $\lambda$. Then we know by induction that $z$ is
divisible by $a_{U}$, say $z=x\cdot a_{U}$ for some $x\in X(A,m+U)$. We let
$K$ denote the kernel of $\chi$. Since $\chi$ does not occur in $U$, the
representation $U$ has trivial $K$-fixed points. So
$\operatorname{res}^{A}_{K}(a_{U})=a_{U|_{K}}$ is a non zero-divisor in
$X(K,\star)$. Since $x\cdot a_{U}=z$ is divisible by $a_{\chi}^{m_{\chi}}$,
part (i) shows that $x$ is divisible by $a_{\chi}^{m_{\chi}}$. So $z=x\cdot
a_{U}$ is divisible by $a_{\chi}^{m_{\chi}}\cdot a_{U}=a_{V}$. ∎

######  Definition 1.7 (Oriented ${\operatorname{el}^{RO}_{2}}$-algebras).

An inverse Thom class of an orientable ${\operatorname{el}^{RO}_{2}}$-algebra
$(X,a)$ is a class $t\in X(C,1-\sigma)$ such that
$\operatorname{res}^{C}_{1}(t)=1$, the multiplicative unit in $X(1,0)$. An
oriented ${\operatorname{el}^{RO}_{2}}$-algebra is a triple $(X,a,t)$
consisting of an orientable ${\operatorname{el}^{RO}_{2}}$-algebra and an
inverse Thom class.

######  Remark 1.8 (The choice of inverse Thom class).

If $(X,a)$ is an orientable ${\operatorname{el}^{RO}_{2}}$-algebra, then in
particular the restriction homomorphism

$\operatorname{res}^{C}_{1}\ :\ X(C,1-\sigma)\ \longrightarrow\ X(1,0)$

is surjective. So an inverse Thom class exists. Moreover, any two inverse Thom
classes differ by an element of the form $a\cdot y$, for a unique $y\in
X(C,1)$; so the set of inverse Thom classes is a torsor over the additive
group $X(C,1)$. In some important cases, the group $X(C,1)$ is trivial, so
that there is a unique inverse Thom class. This holds for example for
equivariant bordism, and for Bredon homology.

The choice of inverse Thom class is essential, and the automorphism group of
an orientable ${\operatorname{el}^{RO}_{2}}$-algebra need not act transitively
on the set of inverse Thom classes. An explicit example is the orientable
${\operatorname{el}^{RO}_{2}}$-algebra $(H\wedge H)^{\sharp}$ arising from the
‘global dual Steenrod algebra’, where $H=H\underline{{\mathbb{F}}}_{2}$ is the
global Eilenberg-MacLane spectrum of the constant Mackey functor. As we plan
to discuss elsewhere, $(H\wedge H)^{\sharp}$ has exactly four inverse Thom
classes, falling into two distinct orbits under the automorphism group of
$(H\wedge H)^{\sharp}$.

Similarly as for the pre-Euler class, we will use the notation

$t_{\lambda}\ =\ \lambda^{*}(t)\ \in\ X(A,1-\lambda)$

for an inverse Thom class $t$ and an $A$-character $\lambda$. The inverse Thom
class of the trivial character is the multiplicative unit 1. And we extend the
inverse Thom classes to general $A$-representations by defining

(1.9) $t_{V}\ =\
\prod_{\lambda\in\operatorname{Hom}(A,C)}t_{\lambda}^{m_{\lambda}}\ ,$

where $m_{\lambda}$ is the multiplicity of the character $\lambda$ in $V$.
These inverse Thom classes are also multiplicative and natural for
restriction, i.e.,

$t_{V}\cdot t_{W}\ =\ t_{V\oplus W}\text{\quad and\quad}\beta^{*}(t_{V})\ =\
t_{\beta^{*}(V)}$

for all homomorphisms $\beta:B\longrightarrow A$ between elementary abelian
2-groups.

######  Proposition 1.10.

Let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra.

  1. (i)

For every subgroup $B$ of $A$, and for all $m\in I_{A}$, the restriction
homomorphism

$\operatorname{res}^{A}_{B}\ :\ X(A,m)\ \longrightarrow\
X(B,\operatorname{res}^{A}_{K}(m))$

is surjective.

  2. (ii)

The $I_{A}$-graded algebra $X(A,\star)$ is generated by its
${\mathbb{Z}}$-graded subalgebra $X(A)_{*}$ and the classes $a_{\lambda}$ and
$t_{\lambda}$ for all nontrivial $A$-characters $\lambda$.

###### Proof.

We prove both statements together by induction over the rank of $A$. The
induction starts when $A$ is the trivial group, in which case there is nothing
to show. Now we let $A$ be a nontrivial elementary abelian 2-group, and we
assume that (i) and (ii) hold for all proper subgroups of $A$.

To prove (i) we may assume that $B$ is a proper subgroup of $A$. We write
$m=k-W$ for an $A$-representation $W$ with $W^{A}=0$. By (ii) for $B$, it
suffices to show that all classes of the form $x\cdot a_{U}\cdot t_{V}$ are in
the image of the restriction homomorphism, whenever $U$ and $V$ are
$B$-representations such that $U\oplus V=W|_{B}$, and $x\in X(B)_{k}$. Because
$W$ is a sum of 1-dimensional $A$-representations, we may choose an
$A$-equivariant decomposition $W=\bar{U}\oplus\bar{V}$ such that
$\bar{U}|_{B}\cong U$ and $\bar{V}|_{B}\cong V$. We also choose a homomorphism
$r:A\longrightarrow B$ that is a retraction to the inclusion. Then

$\operatorname{res}^{A}_{B}(r^{*}(x)\cdot a_{\bar{U}}\cdot t_{\bar{V}})\ =\
x\cdot a_{U}\cdot t_{V}\ ,$

and we have shown (i) for $A$.

For (ii) we also fix a representation grading $m=k-W$, with $W^{A}=0$. We
prove the claim for all classes in $X(A,k-W)$ by induction over the dimension
of $W$. There is nothing to show if $W=0$. Otherwise we write
$W=V\oplus\lambda$ for an $A$-representation $V$ and a nontrivial
$A$-character $\lambda$, with kernel $K$. By part (i), the restriction map
$\operatorname{res}^{A}_{K}:X(A,k-1-V)\longrightarrow X(K,k-1-V|_{K})$ is
surjective. So given $x\in X(A,k-W)$, there is a class $z\in X(A,k-1-V)$ such
that $\operatorname{res}^{A}_{K}(z)=\operatorname{res}^{A}_{K}(x)$. Then

$\operatorname{res}^{A}_{K}(x+t_{\lambda}\cdot z)\ =\
\operatorname{res}^{A}_{K}(x)+\operatorname{res}^{A}_{K}(z)\ =\ 0\ .$

The fundamental exactness property of an orientable
${\operatorname{el}^{RO}_{2}}$-algebra provides a class $y\in X(A,k-V)$ such
that $a_{\lambda}\cdot y=x+t_{\lambda}\cdot z$. By the inductive hypothesis,
the classes $y$ and $z$ are $X(A)_{*}$-linear combinations of products of
$a$-classes and $t$-classes. Hence the same is true for $x=y\cdot
a_{\lambda}+z\cdot t_{\lambda}$, and we have shown (ii). ∎

##  2\. Oriented global ring spectra

In this section we explain how orientable global ring spectra give rise to
orientable ${\operatorname{el}^{RO}_{2}}$-algebras, see Construction 2.4 and
Theorem 2.5, leading to a wealth of interesting examples. For the purpose of
this paper, a global ring spectrum is a commutative monoid object in the
global stable homotopy category of [25, Section 4.4], under the globally
derived smash product [25, (4.2.25)]. So informally speaking, a global ring
spectrum is a consistent collection of $G$-equivariant homotopy ring spectra,
for every compact Lie group $G$.

In this paper, we mostly care about the underlying $G$-spectra of a global
spectrum when $G$ is an elementary abelian 2-group. So we could have worked in
the $\operatorname{el}_{2}$-global stable homotopy category, the Bousfield
localization of the global stable homotopy category that is designed so that
it only ‘sees’ the values of a global spectrum at the global family
$\operatorname{el}_{2}$ of elementary abelian 2-groups. However, since every
$\operatorname{el}_{2}$-global ring spectrum is underlying a global ring
spectrum (in many different ways), working with global ring spectra is no loss
of generality. Moreover, some interesting results, specifically Proposition
2.1 and Theorem 2.3 below, refer to general compact Lie groups.

We recall that a global functor in the sense of [25, Definition 4.2.2] is an
additive functor from the global Burnside category of [25, Construction 4.2.1]
to the category of abelian groups. In more explicit terms, a global functor
specifies values on all compact Lie groups, restriction homomorphisms along
continuous group homomorphisms, and transfers along inclusions of closed
subgroups; these data have to satisfy a short list of explicit relations that
can be found after Theorem 4.2.6 of [25]. The primary invariant of a global
spectrum $E$ is the homotopy group global functor
${\underline{\pi}}_{0}(E)=\\{\pi_{0}^{G}(E)\\}_{G}$ that records the 0-th
equivariant homotopy groups for all compact Lie groups, along with the
restriction and transfer maps that relate them, see [25, Example 4.2.3]. One
of the transfer homomorphisms plays a particularly important role for this
paper, namely the transfer

$\operatorname{tr}_{1}^{C}\ :\ \pi_{0}(E)\ \longrightarrow\ \pi_{0}^{C}(E)$

from the homotopy group of the underlying non-equivariant spectrum to the
$C$-equivariant homotopy group.

In the following proposition, ${\mathbb{S}}$ denotes the global sphere
spectrum, so that ${\underline{\pi}}_{0}({\mathbb{S}})$ is the Burnside ring
global functor, compare [25, Example 4.2.7]. The value of
${\underline{\pi}}_{0}({\mathbb{S}})$ at a compact Lie group $G$ is the
$G$-equivariant stable stem
$\pi_{0}^{G}({\mathbb{S}})=\operatorname{colim}[S^{V},S^{V}]_{*}^{G}$, where
the colimit is over finite-dimensional $G$-subrepresentations of a complete
$G$-universe. When $G$ is finite, $\pi_{0}^{G}({\mathbb{S}})$ is isomorphic to
the Grothendieck ring of isomorphism classes of finite $G$-sets.

For every closed subgroup $H$ of $G$, the ring homomorphism

$\Phi^{H}\ :\ \pi_{0}^{G}({\mathbb{S}})\ \longrightarrow\ {\mathbb{Z}}$

sends the class of a based continuous $G$-map $f:S^{V}\longrightarrow S^{V}$
to the degree of the restriction $f^{H}:S^{V^{H}}\longrightarrow S^{V^{H}}$ to
$H$-fixed points. These homomorphisms are jointly injective, i.e., classes in
$\pi_{0}^{G}({\mathbb{S}})$ are detected by the collection of fixed point
degrees, see for example [25, Theorem 3.3.15].

######  Proposition 2.1.

Let $\langle\operatorname{tr}_{1}^{C}\rangle$ denote the global subfunctor of
${\underline{\pi}}_{0}({\mathbb{S}})$ generated by the transfer
$\operatorname{tr}_{1}^{C}(1)$ in $\pi_{0}^{C}({\mathbb{S}})$. Let $G$ be a
compact Lie group.

  1. (i)

A class $x\in\pi_{0}^{G}({\mathbb{S}})$ belongs to
$\langle\operatorname{tr}_{1}^{C}\rangle(G)$ if and only if for every closed
subgroup $H$ of $G$ with finite Weyl group, the integer $\Phi^{H}(x)$ is even.

  2. (ii)

For every unit $u\in\pi_{0}^{G}({\mathbb{S}})$, the class $u-1$ belongs to
$\langle\operatorname{tr}_{1}^{C}\rangle(G)$.

###### Proof.

(i) As we show in the proof of [25, Proposition 6.1.45], the value
$\langle\operatorname{tr}_{1}^{C}\rangle(G)$ is the subgroup of
$\pi_{0}^{G}({\mathbb{S}})$ generated by $2\cdot\pi_{0}^{G}({\mathbb{S}})$ and
by the transfers $t_{H}^{G}=\operatorname{tr}_{H}^{G}(1_{H})$ for all closed
subgroups $H$ of $G$ whose Weyl group is finite and of even order. The degree
homomorphism $\Phi^{K}:\pi_{0}^{G}({\mathbb{S}})\longrightarrow{\mathbb{Z}}$
is additive, so it takes even values on all classes in
$2\cdot\pi_{0}^{G}({\mathbb{S}})$. If the Weyl group of $H$ in $G$ is finite
of even order, then there is a closed subgroup $L$ of $G$ that contains $H$ as
a normal subgroup of index 2. Then

$\Phi^{K}(t_{H}^{G})\ =\
\Phi^{K}(\operatorname{res}^{G}_{K}(\operatorname{tr}_{L}^{G}(\operatorname{tr}_{H}^{L}(1_{H}))))\
.$

The Weyl group $W_{G}H$ acts freely from the right on the coset space $G/H$,
and hence it acts freely and smoothly on the smooth closed manifold
$(G/H)^{K}$. By [25, Proposition 3.4.2 (ii)], the integer
$\Phi^{K}(t_{H}^{G})$ is divisible by the order of $W_{G}H$, and hence even.
We have thus shown that every class in
$\langle\operatorname{tr}_{1}^{C}\rangle(G)$ has even fixed point degrees.

For the converse we consider a class with even fixed point degrees. By the
previous paragraph, any such class is congruent modulo
$\langle\operatorname{tr}_{1}^{C}\rangle(G)$ to a class of the form

$t_{K_{1}}^{G}+\dots+t_{K_{n}}^{G}$

for pairwise non-conjugate closed subgroups $K_{1},\dots,K_{n}$ of $G$ whose
Weyl groups are finite and of odd order. If the above sum had at least one
summand, we assume without loss of generality that $K_{1}$ is maximal with
respect to subconjugacy among the groups that occur, i.e., $K_{1}$ is not
subconjugate to $K_{i}$ for $i=2,\dots,n$. Then for $i=2,\dots,n$, the fixed
point set $(G/K_{i})^{K_{1}}$ is empty, and thus
$\Phi^{K_{1}}(t_{K_{i}}^{G})=0$. Hence

$\Phi^{K_{1}}(t_{K_{1}}^{G}+\dots+t_{K_{n}}^{G})\ =\
\Phi^{K_{1}}(t_{K_{1}}^{G})\ =\ |W_{G}K_{1}|\ .$

This contradicts the assumption that all fixed point degrees are even. So the
above sum must be empty, i.e., the given class belongs to
$\langle\operatorname{tr}_{1}^{C}\rangle(G)$.

(ii) The degree homomorphism
$\Phi^{H}:\pi_{0}^{G}({\mathbb{S}})\longrightarrow{\mathbb{Z}}$ is
multiplicative, so it sends all units of $\pi_{0}^{G}({\mathbb{S}})$ to
$\\{\pm 1\\}$. Hence $\Phi^{H}(u-1)$ is even, and part (i) proves the claim. ∎

Now we introduce orientable global ring spectra, our main source of examples
for orientable ${\operatorname{el}^{RO}_{2}}$-algebras.

######  Definition 2.2.

A global ring spectrum $E$ is orientable if the class
$\operatorname{tr}_{1}^{C}(1)$ in the group $E_{0}^{C}=\pi_{0}^{C}(E)$ is
trivial.

Some important examples of orientable global ring spectra are the mod 2 global
Eilenberg-MacLane spectrum $H\underline{{\mathbb{F}}}_{2}$, see Example 3.2,
and the global Thom spectrum $\mathbf{mO}$ defined in [25, Example 6.1.24].
These are specific global equivariant forms of the mod 2 Eilenberg-MacLane
spectrum and the classical real bordism spectrum. The global ring spectrum
$H\underline{{\mathbb{F}}}_{2}$ represents Bredon homology with constant
coefficients, and the global ring spectrum $\mathbf{mO}$ realizes the
orientable ${\operatorname{el}^{RO}_{2}}$-algebra $\mathcal{N}$ of equivariant
bordism. We return to these examples in much detail in Sections 3 and 6,
respectively. Further examples that we discuss below are the global Thom ring
spectrum $\mathbf{MO}$ defined in [25, Example 6.1.7] that represents stable
equivariant bordism, see Corollary 8.6 and the remarks immediately thereafter;
and the global Borel ring spectrum associated to the mod 2 Eilenberg-MacLane
spectrum, see Theorem 8.8. For these four examples, we establish universal
characterizations of the associated oriented
${\operatorname{el}^{RO}_{2}}$-algebra, see Theorem 3.5, Theorem 7.14,
Corollary 8.6 and Theorem 8.8, respectively. The global smash product of an
orientable global ring spectrum with any other global ring spectrum is again
orientable, so this yields many further examples.

The next theorem shows that orientability of a global ring spectrum has rather
strong consequences for the homotopy group global functor.

######  Theorem 2.3.

Let $E$ be an orientable global ring spectrum, and let $G$ be a compact Lie
group.

  1. (i)

All $G$-equivariant homotopy groups of $E$ are ${\mathbb{F}}_{2}$-vector
spaces.

  2. (ii)

The ring homomorphism
$\eta_{*}:\pi_{0}^{G}({\mathbb{S}})\longrightarrow\pi_{0}^{G}(E)=E_{0}^{G}$
induced by the unit morphism $\eta:{\mathbb{S}}\longrightarrow E$ sends all
units of $\pi_{0}^{G}({\mathbb{S}})$ to 1.

  3. (iii)

For every $G$-representation $V$ and every $G$-equivariant based homotopy
self-equivalence $\psi:S^{V}\longrightarrow S^{V}$, the automorphism
$E_{*}^{G}(S^{\psi})$ of $E_{*}^{G}(S^{V})$ is the identity.

###### Proof.

(i) The vanishing of the class $\operatorname{tr}_{1}^{C}(1)$ yields

$2\ =\ \operatorname{res}^{C}_{1}(\operatorname{tr}_{1}^{C}(1))\ =\ 0$

in $\pi_{0}(E)$. Since all equivariant homotopy groups of $E$ are modules over
$\pi_{0}(E)$ via inflation, they are all ${\mathbb{F}}_{2}$-vector spaces.

(ii) As $G$ varies over all compact Lie groups, the homomorphisms
$\eta_{*}:\pi_{0}^{G}({\mathbb{S}})\longrightarrow\pi_{0}^{G}(E)$ form a
morphism of global functors
$\eta_{*}:{\underline{\pi}}_{0}({\mathbb{S}})\longrightarrow{\underline{\pi}}_{0}(E)$.
Because $\operatorname{tr}_{1}^{C}(1)=0$ in $\pi_{0}^{C}(E)$, the global
subfunctor $\langle\operatorname{tr}_{1}^{C}\rangle$ of
${\underline{\pi}}_{0}({\mathbb{S}})$ maps to 0 under the morphism $\eta_{*}$.
Hence by Proposition 2.1 (ii), the homomorphism $\eta_{*}$ sends all units in
$\pi_{0}^{G}({\mathbb{S}})$ to 1.

(iii) The automorphism $E_{*}^{G}(S^{\psi})$ of $E_{*}^{G}(S^{V})$ coincides
with multiplication by the image of $[\psi]\in\pi_{0}^{G}({\mathbb{S}})$ under
the ring homomorphism
$\eta_{*}:\pi_{0}^{G}({\mathbb{S}})\longrightarrow\pi_{0}^{G}(E)$. Since
$\psi$ is a homotopy self-equivalence, the class $[\psi]$ is a unit in
$\pi_{0}^{G}({\mathbb{S}})$, so part (ii) proves the claim. ∎

######  Construction 2.4 (The ${\operatorname{el}^{RO}_{2}}$-algebra of an
orientable global ring spectrum).

We let $E$ be an orientable global ring spectrum. We will now define an
associated ${\operatorname{el}^{RO}_{2}}$-algebra $E^{\sharp}$. We write any
given $m\in I_{A}$ as $m=k-V$ for an integer $k$ and an $A$-representation $V$
with trivial fixed points. Then $k$ is unique, and $V$ is unique up to
isomorphism. We set

$E^{\sharp}(A,m)\ =\ E^{\sharp}(A,k-V)\ =\ E_{k}^{A}(S^{V})\ .$

A key point is that this assignment is independent up to preferred isomorphism
of the choice of $V$. Indeed, an isomorphism of $A$-representations
$\psi:V\longrightarrow W$ induces an equivariant homeomorphism
$S^{\psi}:S^{V}\longrightarrow S^{W}$, and hence an isomorphism of equivariant
homotopy groups

$E_{k}^{A}(S^{\psi})\ :\ E_{k}^{A}(S^{V})\ \longrightarrow\ E_{k}^{A}(S^{W})\
.$

Theorem 2.3 (iii) guarantees that this isomorphism does not depend on the
choice of $\psi$.

The restriction map along a morphism
$\alpha:(B,\alpha^{*}(m))\longrightarrow(A,m)$ in
${\operatorname{el}^{RO}_{2}}$ is defined as follows. We choose an
$A$-representation $V$ and a $B$-representation $W$ without fixed points so
that $m=k-V$ and $\alpha^{*}(m)=l-W$ for integers $k\geq l$. There is then an
isomorphism of $A$-representations

$\psi\ :\ \alpha^{*}(V)\ \xrightarrow{\cong}\ W\oplus{\mathbb{R}}^{k-l}\ .$

We define

$\alpha^{*}\ :\ E^{\sharp}(A,m)\ \longrightarrow\ E^{\sharp}(B,\alpha^{*}(m))$

as the composite

$\displaystyle E_{k}^{A}(S^{V})\ $ $\displaystyle\xrightarrow{\alpha^{*}}\
E_{k}^{B}(S^{\alpha^{*}(V)})\ \xrightarrow{E_{k}^{B}(S^{\psi})}\
E_{k}^{B}(S^{W\oplus{\mathbb{R}}^{k-l}})\ \xleftarrow[\cong]{-\wedge S^{k-l}}\
E_{l}^{B}(S^{W})\ .$

The first map is the restriction homomorphism of equivariant homotopy groups
[25, Construction 3.1.15], and the final isomorphism pointing backwards is the
suspension isomorphism. Theorem 2.3 (iii) guarantees that $\alpha^{*}$ does
not depend on the choice of $\psi$. The multiplication pairings are defined as
the composite

$\displaystyle E^{\sharp}(A,m)\times E^{\sharp}(B,n)\ $ $\displaystyle=\
E_{k}^{A}(S^{V})\times E_{l}^{B}(S^{W})$ $\displaystyle\xrightarrow{\ \wedge\
}\ \pi_{k+l}^{A\times B}(E\wedge S^{V}\wedge E\wedge S^{W})$
$\displaystyle\xrightarrow{\ \mu_{*}\ }\ \pi_{k+l}^{A\times B}(E\wedge
S^{V\oplus W})\ =\ E^{\sharp}(A\times B,m\oplus n)\ .$

Here we have exploited that $m+n=(k+l)-(V\oplus W)$ in $I_{A\times B}$.

The fixed point inclusion $S^{0}\longrightarrow S^{\sigma}$ defines an element
$a\in\pi_{0}^{C}({\mathbb{S}}\wedge S^{\sigma})$ that is sometimes called the
(pre-)Euler class of the sign representation; we abuse notation and use the
same symbol for the image

$a\in E_{0}^{C}(S^{\sigma})\ =\ E^{\sharp}(C,-\sigma)$

under the unit morphism ${\mathbb{S}}\longrightarrow E$.

######  Theorem 2.5.

For every orientable global ring spectrum $E$, the pair $(E^{\sharp},a)$ is an
orientable ${\operatorname{el}^{RO}_{2}}$-algebra.

###### Proof.

We exploit that the sequence of equivariant homotopy groups

$E_{1}^{C}(S^{\sigma})\ \xrightarrow{\operatorname{res}^{C}_{e}}\
E_{1}(S^{1})\ \xrightarrow{\ \operatorname{tr}_{1}^{C}\ }\ E_{1}^{C}(S^{1})$

is exact. Because $E$ is orientable, we have
$\operatorname{tr}^{C}_{1}(1\wedge S^{1})=\operatorname{tr}^{C}_{1}(1)\wedge
S^{1}=0$, and we can choose a class $t\in E_{1}^{C}(S^{\sigma})$ that
restricts to $1\wedge S^{1}$. We use $t$ to define classes $t_{V}\in
E_{d}^{A}(S^{V})$ as in (1.9), for all representations $V$ of elementary
abelian 2-groups $A$, where $d=\dim(V)$. We claim that the following three
properties hold for all $A$-representations $W$:

  1. (i)

For every subgroup $B$ of $A$, the restriction homomorphism

$\operatorname{res}^{A}_{B}\ :\ E_{k}^{A}(S^{W})\ \longrightarrow\
E_{k}^{B}(S^{W|_{B}})$

is surjective.

  2. (ii)

Suppose that $W=V\oplus\lambda$ for a nontrivial $A$-character $\lambda$ with
kernel $K$. Then the following sequence is exact:

$0\ \longrightarrow\ E_{k}^{A}(S^{V})\ \xrightarrow{\ \cdot a_{\lambda}}\
E_{k}^{A}(S^{V\oplus\lambda})\ \xrightarrow{\operatorname{res}^{A}_{K}}\
E_{k-1}^{K}(S^{V|_{K}})\ \longrightarrow\ 0$

  3. (iii)

The graded ${\mathbb{F}}_{2}$-vector space $E_{*}^{A}(S^{W})$ is generated as
a graded $E_{*}^{A}$-module by the classes $a_{U}\cdot t_{V}$ for all
$A$-representations $U$ and $V$ such that $U\oplus V=W$.

Part (ii) is the exactness property that shows that $(E^{\sharp},a)$ is an
orientable ${\operatorname{el}^{RO}_{2}}$-algebra.

The argument is similar as in the proof of Proposition 1.10: we prove all
three statements together by induction over the rank of $A$. The induction
starts when $A$ is the trivial group, in which case there is nothing to show.
Now we let $A$ be a nontrivial elementary abelian 2-group, and we assume that
parts (i)–(iii) hold for all proper subgroups of $A$.

We start by proving (i), where we may assume that $B$ is a proper subgroup of
$A$. By part (iii) for $B$, it suffices to show that all classes of the form
$x\cdot a_{U}\cdot t_{V}$ are in the image of the restriction homomorphism,
whenever $U$ and $V$ are $B$-representations such that $U\oplus V=W|_{B}$, and
$x$ is a homogeneous element of $E_{*}^{B}$. Because $W$ is a sum of
1-dimensional $A$-representations, we may choose an $A$-equivariant
decomposition $W=\bar{U}\oplus\bar{V}$ such that
$\operatorname{res}^{A}_{B}(\bar{U})\cong U$ and
$\operatorname{res}^{A}_{B}(\bar{V})\cong V$. We also choose a homomorphism
$r:A\longrightarrow B$ that is a retraction to the inclusion. Then

$\operatorname{res}^{A}_{B}(r^{*}(x)\cdot a_{\bar{U}}\cdot t_{\bar{V}})\ =\
x\cdot a_{U}\cdot t_{V}\ ,$

and we have shown (i) for $A$. To prove (ii) we smash the cofiber sequence of
based $A$-spaces

$A/K_{+}\ \xrightarrow{\quad}\ S^{0}\ \xrightarrow{\quad}\ S^{\lambda}\
\longrightarrow\ A/K_{+}\wedge S^{1}$

with $S^{V}$ and apply $A$-equivariant $E$-homology, yielding a long exact
sequence:

$\displaystyle\dots\ \longrightarrow\ E_{k}^{A}(S^{V})\ \xrightarrow{\ \cdot
a_{\lambda}}\ E_{k}^{A}(S^{V\oplus\lambda})\ \xrightarrow{\ \partial\ }\
E_{k-1}^{A}(S^{V}\wedge A/K_{+})\ \longrightarrow\ \dots$

The Wirthmüller isomorphism identifies the group $E_{k-1}^{A}(S^{V}\wedge
A/K_{+})$ with $E_{k-1}^{K}(S^{V|_{K}})$. Under this identification, the
boundary map $\partial$ becomes the restriction homomorphism
$\operatorname{res}^{A}_{K}:E_{k}^{A}(S^{V\oplus\lambda})\longrightarrow
E_{k-1}^{K}(S^{V|_{K}})$, which is surjective by (i). So the long exact
sequence decomposes into short exact sequences, showing (ii).

We prove (iii) by induction on $\dim(W)-\dim(W^{A})$. If $A$ acts trivially on
$W$, the $E_{*}^{A}$-module $E_{*}^{A}(S^{W})$ is free of rank 1 with basis
the class $t_{W}$, so the claim holds. If $W$ acts nontrivially on $W$, we
write $W=V\oplus\lambda$ for an $A$-representation $V$ and a nontrivial
$A$-character $\lambda$, with kernel $K$. By part (i), the restriction map
$\operatorname{res}^{A}_{K}:E_{m-1}^{A}(S^{V})\longrightarrow
E_{m-1}^{K}(S^{V|_{K}})$ is surjective. So given $x\in E_{k}^{A}(S^{W})$,
there is a class $z\in E_{m-1}^{A}(S^{V})$ such that
$\operatorname{res}^{A}_{K}(z)=\operatorname{res}^{A}_{K}(x)$. Then

$\operatorname{res}^{A}_{K}(x+z\cdot t_{\lambda})\ =\
\operatorname{res}^{A}_{K}(x)+\operatorname{res}^{A}_{K}(z)\ =\ 0\ .$

Part (ii) provides a class $y\in E_{k}^{A}(S^{V})$ such that $y\cdot
a_{\lambda}=x+z\cdot t_{\lambda}$. By the inductive hypothesis, the classes
$y$ and $z$ are $E_{*}^{A}$-linear combinations of products of $a$-classes and
$t$-classes. Hence the same is true for $x=y\cdot a_{\lambda}+z\cdot
t_{\lambda}$, and we have shown (iii). ∎

##  3\. The universal property of Bredon homology

In this section we discuss Bredon ${\operatorname{el}^{RO}_{2}}$-algebras,
i.e., orientable ${\operatorname{el}^{RO}_{2}}$-algebras whose
${\mathbb{Z}}$-graded parts consist only of a copy of ${\mathbb{F}}_{2}$ in
dimension zero, see Definition 3.1. As the name suggests, Bredon homology of
representation spheres with constant mod 2 coefficients is an example of a
Bredon ${\operatorname{el}^{RO}_{2}}$-algebra. Our main result, Theorem 3.5,
exhibits a universal property: every Bredon
${\operatorname{el}^{RO}_{2}}$-algebra is initial among additive oriented
${\operatorname{el}^{RO}_{2}}$-algebras. It follows that Bredon
${\operatorname{el}^{RO}_{2}}$-algebras are unique up to unique isomorphism, a
fact that is not entirely obvious, at least to the authors, from the
definition.

######  Definition 3.1.

A Bredon ${\operatorname{el}^{RO}_{2}}$-algebra is an orientable
${\operatorname{el}^{RO}_{2}}$-algebra $H$ such that for every elementary
abelian 2-group $A$ and all $k\in{\mathbb{Z}}$ we have

$H(A)_{k}\ =\ H(A,k)\ =\ \begin{cases}{\mathbb{F}}_{2}&\text{ for $k=0$,
and}\\\ \ 0&\text{ for $k\neq 0$.}\end{cases}$

As the following example shows, Bredon ${\operatorname{el}^{RO}_{2}}$-algebras
exist. Theorem 3.5 below proves that Bredon
${\operatorname{el}^{RO}_{2}}$-algebras are unique up to unique isomorphism.

For every ${\operatorname{el}^{RO}_{2}}$-algebra $X$, any two choices of pre-
Euler classes differ by multiplication with a unit in $X(C,0)$. Furthermore,
as we discussed in Remark 1.8, the difference between two choices of inverse
Thom classes is an element of the form $a\cdot y$, for $y\in X(C,1)$. If $H$
is a Bredon ${\operatorname{el}^{RO}_{2}}$-algebra, then
$H(C,0)^{\times}={\mathbb{F}}_{2}^{\times}=\\{1\\}$ and $H(C,1)=0$. Therefore,
the pre-Euler class and the inverse Thom class of a Bredon
${\operatorname{el}^{RO}_{2}}$-algebra are unique.

######  Example 3.2 (The global Eilenberg-MacLane spectrum).

We let $H\underline{{\mathbb{F}}}_{2}$ denote the classical commutative
orthogonal ring spectrum model for the Eilenberg-MacLane spectrum for the
field ${\mathbb{F}}_{2}$, as defined for example in [25, Construction 5.3.8].
By [25, Proposition 5.3.9], its restriction to finite groups is an Eilenberg-
MacLane ring spectrum for the constant global functor with value
${\mathbb{F}}_{2}$. So $H\underline{{\mathbb{F}}}_{2}$ represents Bredon
homology and cohomology with coefficients in the constant Mackey functor
${\mathbb{F}}_{2}$.

Because the transfer $\operatorname{tr}_{1}^{C}(1)$ is trivial in Bredon
homology with constant mod 2 coefficients, $H\underline{{\mathbb{F}}}_{2}$ is
an orientable global ring spectrum. By Theorem 2.5, the pair
$((H\underline{{\mathbb{F}}}_{2})^{\sharp},a)$ is an orientable
${\operatorname{el}^{RO}_{2}}$-algebra, and hence a Bredon
${\operatorname{el}^{RO}_{2}}$-algebra in the sense of Definition 3.1.

######  Definition 3.3 (Additive oriented
${\operatorname{el}^{RO}_{2}}$-algebras).

An oriented ${\operatorname{el}^{RO}_{2}}$-algebra $(X,a,t)$ is additive if
the class

$a_{1}t_{\mu}t_{2}\ +\ t_{1}a_{\mu}t_{2}\ +\ t_{1}t_{\mu}a_{2}$

in $X(C\times C,2-(p_{1}\oplus\mu\oplus p_{2}))$ is trivial. Here
$p_{1},p_{2},\mu:C\times C\longrightarrow C$ are the three nontrivial
characters of the group $C\times C$, and we abbreviate $a_{p_{i}}$ to $a_{i}$,
and $t_{p_{i}}$ to $t_{i}$.

The term ‘additive’ in the previous definition is motivated by the fact that
the 2-torsion formal group law associated to an additive
${\operatorname{el}^{RO}_{2}}$-algebra is the additive formal group law, see
Example 4.8 below.

######  Example 3.4 (Bredon ${\operatorname{el}^{RO}_{2}}$-algebras are
additive).

We let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra. The
class $a_{1}t_{\mu}t_{2}+t_{1}a_{\mu}t_{2}+t_{1}t_{\mu}a_{2}$ in $X(C\times
C,2-(p_{1}\oplus\mu\oplus p_{2}))$ is in the kernel of all three restriction
maps from $C\times C$ to $C$. By the fundamental exactness property, this
class is divisible by each of $a_{1}$, $a_{\mu}$ and $a_{2}$. Proposition 1.6
(ii) then shows that the class is divisible by the product
$a_{1}a_{\mu}a_{2}$. So there is a class $z\in X(C\times C)_{2}$ such that

$z\cdot a_{1}a_{\mu}a_{2}\ =\ a_{1}t_{\mu}t_{2}\ +\ t_{1}a_{\mu}t_{2}\ +\
t_{1}t_{\mu}a_{2}\ .$

If the group $X(C\times C)_{2}$ is trivial, then the class $z$ necessarily
vanishes, and so $(X,a,t)$ is additive. This is the case in particular for all
Bredon ${\operatorname{el}^{RO}_{2}}$-algebras.

The following main result of this section is the fact that every Bredon
${\operatorname{el}^{RO}_{2}}$-algebra is initial among additive
${\operatorname{el}^{RO}_{2}}$-algebras. In the proof we will start using the
notation

$A^{\circ}\ =\ \\{\lambda\colon A\longrightarrow C\ |\ \text{$\lambda$ is
surjective}\\}$

for the set of nontrivial characters of an elementary abelian 2-group $A$.

######  Theorem 3.5 (Universal property of Bredon homology).

Let $H$ be a Bredon ${\operatorname{el}^{RO}_{2}}$-algebra, and let $X$ be an
orientable ${\operatorname{el}^{RO}_{2}}$-algebra.

  1. (i)

Evaluation at the unique inverse Thom class of $H$ is a bijection between the
set of morphisms of orientable ${\operatorname{el}^{RO}_{2}}$-algebras from
$H$ to $X$, and the set of those inverse Thom classes $t$ of $X$ such that
$(X,a,t)$ is additive.

  2. (ii)

Any two Bredon ${\operatorname{el}^{RO}_{2}}$-algebras are uniquely
isomorphic.

###### Proof.

(i) The unique orientation and inverse Thom class make $(H,a,t)$ additive, so
every morphism of orientable ${\operatorname{el}^{RO}_{2}}$-algebras sends $t$
to an inverse Thom class of an additive orientation.

Now we prove the uniqueness. Proposition 1.10 (ii) shows that $H(A,\star)$ is
generated as an $I_{A}$-graded ${\mathbb{F}}_{2}$-algebra by the classes
$a_{\lambda}$ and $t_{\lambda}$ for all nontrivial $A$-characters, where $t$
is the unique inverse Thom class; so every morphism $f:H\longrightarrow X$ of
oriented ${\operatorname{el}^{RO}_{2}}$-algebras is determined by its effect
on the pre-Euler class $a$ and the inverse Thom class $t$.

Now we show the existence of the morphism for the specific example
$H^{\sharp}=(H\underline{{\mathbb{F}}}_{2})^{\sharp}$, the Bredon
${\operatorname{el}^{RO}_{2}}$-algebra given by the Bredon homology of
representation spheres, see Example 3.2. We will exploit the presentation of
the rings $H^{\sharp}(A,\star)$ obtained in [18, Theorem 2.5]. We let
$(\bar{a},\bar{t})$ be an additive orientation of $X$. We let $T\subset
A^{\circ}$ be any set of nontrivial $A$-characters whose product
$\prod_{\lambda\in T}\lambda$ is the trivial character. We claim that the
class

$r(T)\ =\ \sum_{\lambda\in T}\bar{a}_{\lambda}\cdot\big{(}\prod_{\mu\in
T\setminus\\{\lambda\\}}\bar{t}_{\mu}\big{)}$

in $X(A,|T|-1-\bigoplus_{\lambda\in T}\lambda)$ equals $0$. We prove this by
induction over the cardinality of $T$. A non-empty subset of $A^{\circ}$ whose
elements multiply to 1 must have at least 3 elements, so we start the
induction for $T=\\{\alpha,\beta,\gamma\\}$ with
$\alpha\cdot\beta\cdot\gamma=1$. Then

$\displaystyle r(\\{\alpha,\beta,\gamma\\})\ $ $\displaystyle=\
\bar{a}_{\alpha}\bar{t}_{\beta}\bar{t}_{\gamma}+\bar{t}_{\alpha}\bar{a}_{\beta}\bar{t}_{\gamma}+\bar{t}_{\alpha}\bar{t}_{\beta}\bar{a}_{\gamma}$
$\displaystyle=\
(\alpha,\gamma)^{*}(\bar{a}_{1}\bar{t}_{\mu}\bar{t}_{2}+\bar{t}_{1}\bar{a}_{\mu}\bar{t}_{2}+\bar{t}_{1}\bar{t}_{\mu}\bar{a}_{2})\
=\ 0$

by the additivity hypothesis, where $(\alpha,\gamma)\colon A\longrightarrow
C\times C$.

Now we suppose that the set $T$ has at least 4 elements. We pick two distinct
elements $\alpha\neq\beta$ from $T$, and we set $\gamma=\alpha\cdot\beta$. If
$\gamma\in T$, then the sets $T\setminus\\{\alpha,\beta,\gamma\\}$ and
$\\{\alpha,\beta,\gamma\\}$ both have fewer elements than $T$, and for both
the product of their elements is the trivial character. So
$r(T\setminus\\{\alpha,\beta,\gamma\\})=0$ and
$r(\\{\alpha,\beta,\gamma\\})=0$ by induction. Thus

$r(T)\ =\ \bar{t}_{\alpha}\bar{t}_{\beta}\bar{t}_{\gamma}\cdot
r(T\setminus\\{\alpha,\beta,\gamma\\})\ +\
r(\\{\alpha,\beta,\gamma\\})\cdot\\!\prod_{\mu\in
T\setminus\\{\alpha,\beta,\gamma\\}}\bar{t}_{\mu}\ =\ 0\ .$

If $\gamma\not\in T$, then the sets
$(T\setminus\\{\alpha,\beta\\})\cup\\{\gamma\\}$ and
$\\{\alpha,\beta,\gamma\\}$ both have fewer elements than $T$, and for both
the product of their elements is the trivial character. So
$r((T\setminus\\{\alpha,\beta\\})\cup\\{\gamma\\})=0$ and
$r(\\{\alpha,\beta,\gamma\\})=0$ by induction. Thus

$\bar{a}_{\gamma}\cdot r(T)\ =\
(\bar{a}_{\alpha}\bar{t}_{\beta}+\bar{t}_{\alpha}\bar{a}_{\beta})\cdot
r((T\setminus\\{\alpha,\beta\\})\cup\\{\gamma\\})\ +\
r(\\{\alpha,\beta,\gamma\\})\cdot r(T\setminus\\{\alpha,\beta\\})\ =\ 0\ .$

Since multiplication by $\bar{a}_{\gamma}$ is injective, we conclude that
$r(T)=0$. This completes the inductive step, and hence the proof of the claim.

By [18, Theorem 2.5], the $I_{A}$-graded ring $H^{\sharp}(A,\star)$ is
generated by the classes $a_{\lambda}$ and $t_{\lambda}$ for all nontrivial
$A$-characters $\lambda$, and the ideal of relations between these classes is
generated by the polynomials $r(T)$ as $T$ runs through all minimally
dependent subsets of $A^{\circ}$. Minimally dependent sets of nontrivial
characters in particular have the property that their elements multiply to the
trivial character. By the previous claim, all generating relations map to 0 in
$X(A,\star)$. So there is a unique morphism of $I_{A}$-graded
${\mathbb{F}}_{2}$-algebras

$f(A,\star)\ :\ H^{\sharp}(A,\star)\ \longrightarrow\ X(A,\star)$

such that $f(A,-\lambda)(a_{\lambda})=\bar{a}_{\lambda}$ and
$f(A,1-\lambda)(t_{\lambda})=\bar{t}_{\lambda}$ for all $\lambda\in
A^{\circ}$. For varying $A$, these homomorphisms are compatible with
restriction along group homomorphisms, simply because this holds for $a$\- and
$t$-classes and these generate $H^{\sharp}(A,\star)$ as an
${\mathbb{F}}_{2}$-algebra. We have thus constructed the desired morphism
$f:H^{\sharp}\longrightarrow X$.

(ii) We let $H$ be any Bredon ${\operatorname{el}^{RO}_{2}}$-algebra. Then $H$
has a unique orientation, which is moreover additive. So part (i) provides a
unique morphism of orientable ${\operatorname{el}^{RO}_{2}}$-algebras $f\colon
H^{\sharp}\longrightarrow H$ that matches the orientations. This morphism is
necessarily an isomorphism in integer degrees, and hence an isomorphism of
orientable ${\operatorname{el}^{RO}_{2}}$-algebras, see Proposition 1.5. ∎

##  4\. The associated 2-torsion formal group law

As we shall now explain, the underlying ${\mathbb{F}}_{2}$-algebra of every
oriented ${\operatorname{el}^{RO}_{2}}$-algebra $(X,a,t)$ naturally comes with
a 2-torsion formal group law, obtained roughly speaking by expanding the Euler
class $e_{\mu}$ of the multiplication $\mu:C\times C\longrightarrow C$ in
terms of the Euler classes of the two projections. As we will see later, a lot
of the structure of $(X,a,t)$ can be recovered from this underlying 2-torsion
formal group law and the geometric fixed point rings of $X$.

######  Construction 4.1 (Localization away from $t$).

We define the localization away from $t$ of an oriented
${\operatorname{el}^{RO}_{2}}$-algebra $(X,a,t)$ by inverting all inverse Thom
classes. For every elementary abelian 2-group $A$, we let $(t^{-1}X)(A)$ be
the integer graded part of the localization of the $I_{A}$-graded ring
$X(A,\star)$ obtained by inverting all classes of the form $t_{\lambda}$ for
all nontrivial $A$-characters $\lambda$. Homogeneous elements of degree $k$ of
$(t^{-1}X)(A)$ are thus fractions of the form $x/t_{V}$ for some
$A$-representation $V$, and some $x\in X(A,k+|V|-V)$. These fractions then
satisfy

$x/t_{V}\ =\ (x\cdot t_{W})/t_{V\oplus W}\text{\qquad and\qquad}x/t_{V}\cdot
y/t_{W}\ =\ (x\cdot y)/t_{V\oplus W}\ .$

The localizations away from $t$ come with restriction homomorphisms
$\alpha^{*}:(t^{-1}X)(A)\longrightarrow(t^{-1}X)(B)$ that are contravariantly
functorial for group homomorphisms $\alpha:B\longrightarrow A$, and defined by
the formula

$\alpha^{*}(x/t_{V})\ =\ \alpha^{*}(x)/t_{\alpha^{*}(V)}\ .$

######  Definition 4.2 (Euler classes).

Let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra. The Euler
class of a nontrivial character $\lambda$ of an elementary abelian 2-group $A$
is the class

$e_{\lambda}\ =\ a_{\lambda}/t_{\lambda}\ =\ \lambda^{*}(a/t)\ \in\
(t^{-1}X)(A)_{-1}$

of degree $-1$ in the localization away from $t$ of $X$.

For $i=1,\dots,n$, we let $p_{i}:C^{n}\longrightarrow C$ be the projection to
the $i$-th factor. We write

$e_{i}\ =\ e_{p_{i}}\ =\ p_{i}^{*}(a/t)\ \in\ (t^{-1}X)(C^{n})_{-1}$

for the Euler class of $p_{i}$.

######  Proposition 4.3.

Let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra and $n\geq
0$. The Euler classes $e_{1},\dots,e_{n}$ form a regular sequence in the
graded ring $(t^{-1}X)(C^{n})$ that generates the kernel of the restriction
homomorphism

$\operatorname{res}^{C^{n}}_{1}\ :\ (t^{-1}X)(C^{n})\ \longrightarrow\ X(1)\
.$

###### Proof.

We argue by induction over $n$, beginning with $n=0$ where there is nothing to
show. Now we assume $n\geq 1$. Because localization is exact, the sequence

$0\ \longrightarrow\ (t^{-1}X)(C^{n})_{*+1}\ \xrightarrow{e_{1}\cdot-}\
(t^{-1}X)(C^{n})_{*}\ \xrightarrow{\ j^{*}\ }\ (t^{-1}X)(C^{n-1})_{*}\
\longrightarrow\ 0$

is exact, where $j=(1,-):C^{n-1}\longrightarrow C^{n}$ is the embedding as the
last $n-1$ factors. So $e_{1}$ is a non zero-divisor, and the restriction
homomorphism $j^{*}$ identifies $(t^{-1}X)(C^{n})/(e_{1})$ with
$(t^{-1}X)(C^{n-1})$. By induction hypothesis, the images of the classes
$e_{1},\dots,e_{n-1}$ are a regular sequence in $(t^{-1}X)(C^{n-1})$ that
generates the augmentation ideal. Since $j^{*}(e_{i})=e_{i-1}$ for $2\leq
i\leq n$, this completes the inductive step. ∎

A key tool in our arguments will involve expansions of elements of certain
rings as power series in a distinguished regular element. We will now set up
some theory around this in a systematic fashion. In the following we will use
the term graded ring as short hand for ‘commutative ${\mathbb{Z}}$-graded
ring’. The next proposition is well-known; since it is central for various of
our arguments, we take the time to spell it out as a point of reference.

######  Proposition 4.4.

Let $(e_{1},\dots,e_{n})$ be a regular sequence of homogeneous elements of a
graded ring $R$. Let $s:S\longrightarrow R$ be a graded ring homomorphism
whose composite with the projection $R\longrightarrow R/(e_{1},\dots,e_{n})$
is an isomorphism.

  1. (i)

The completion of $R$ at the ideal $(e_{1},\dots,e_{n})$ is a power series
algebra over $S$ in the elements $e_{1},\dots,e_{n}$.

  2. (ii)

Let $T$ be a graded ring that is complete with respect to a homogeneous ideal
$J$. Then the map

$\displaystyle\\{\rho\in\text{\em
grRing}(R,T)\colon\rho(e_{1},\dots,e_{n})\subset J\\}\ $
$\displaystyle\longrightarrow\ \text{\em grRing}(S,T)\times J^{n}$
$\displaystyle\rho\qquad$ $\displaystyle\longmapsto\ (\rho
s,\rho(e_{1}),\dots,\rho(e_{n}))$

is bijective.

###### Proof.

We write $I=(e_{1},\dots,e_{n})$ for the ideal generated by the regular
sequence. It is a classical fact that because of the regularity, the unique
$R/I$-algebra homomorphism

$(R/I)[x_{1},\dots,x_{n}]\ \longrightarrow\ \bigoplus_{k\geq 0}I^{k}/I^{k+1}$

that sends $x_{i}$ to $e_{i}+I^{2}$ is an isomorphism, see for example [8,
§9.7, Théorème 1]. In particular, $I^{k}/I^{k+1}$ is a free $R/I$-module on
the residue classes of the monomials of exact degree $k$ in
$e_{1},\dots,e_{n}$. Thus by induction on $k$, the underlying $S$-module of
$R/I^{k+1}$ is free on the residue classes of the monomials of degree at most
$k$. So $R/I^{k+1}$ is a truncated polynomial algebra over $S$, and the
completion of $R$ at $I$ is a power series algebra over $S$ on the classes
$e_{1},\dots,e_{n}$, giving (i). This power series algebra is free when it
comes to morphisms with complete targets, yielding (ii). ∎

######  Definition 4.5.

Let $(e_{1},\dots,e_{n})$ be a regular sequence of homogeneous elements of a
commutative graded ring $R$, and let $s:S\longrightarrow R$ be a ring
homomorphism whose composite with the projection $R\longrightarrow
R/(e_{1},\dots,e_{n})$ is an isomorphism. The power series expansion of $R$ at
$e_{1},\dots,e_{n}$ is the unique morphism $d:R\longrightarrow S\llbracket
e_{1},\dots,e_{n}\rrbracket$ of graded $S$-algebras that sends $e_{i}$ to
$e_{i}$ for all $1\leq i\leq n$.

######  Construction 4.6.

We let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra and
$n\geq 0$. The inflation homomorphism
$p^{*}:X(1)\longrightarrow(t^{-1}X)(C^{n})$ is a section to the restriction
homomorphism $\operatorname{res}^{C^{n}}_{1}$. Since the kernel of
$\operatorname{res}^{C^{n}}_{1}$ is generated by the regular sequence of Euler
classes $e_{1},\dots,e_{n}$, Proposition 4.4 provides a unique graded
$X(1)$-algebra homomorphism

$d^{n}\ :\ (t^{-1}X)(C^{n})\ \longrightarrow\ X(1)\llbracket
e_{1},\dots,e_{n}\rrbracket$

that sends $e_{i}$ to $e_{i}$ for all $1\leq i\leq n$, the power series
expansion in the Euler classes $e_{1},\dots,e_{n}$.

We recall that a 2-torsion formal group law over a graded commutative ring $R$
is a power series $F(x,y)\in R\llbracket x,y\rrbracket$ that is homogeneous of
degree $-1$ with respect to the grading $\deg(x)=\deg(y)=-1$, and that
satisfies

$F(x,0)=x,\quad F(x,y)=F(y,x),\quad F(x,F(y,z))=F(F(x,y),z)\text{\quad
and\quad}F(x,x)=0\ .$

In the next theorem and in what follows, we shall write $\mu:C\times
C\longrightarrow C$ for the multiplication homomorphism.

######  Theorem 4.7.

Let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra. The image
of the Euler class $e_{\mu}$ under the power series expansion

$d^{2}\ :\ (t^{-1}X)(C\times C)\ \longrightarrow\ X(1)\llbracket
e_{1},e_{2}\rrbracket$

is a 2-torsion formal group law over the graded ring $X(1)$, the formal group
law of $(X,a,t)$.

###### Proof.

We let $\alpha:C^{n}\longrightarrow C^{2}$ be a group homomorphism, for some
$n\geq 1$. We claim that the following diagram of graded $X(1)$-algebras
commutes:

$\textstyle{(t^{-1}X)(C^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{2}}$$\scriptstyle{\alpha^{*}}$$\textstyle{(t^{-1}X)(C^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{n}}$$\textstyle{X(1)\llbracket
e_{1},e_{2}\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\natural}}$$\textstyle{X(1)\llbracket
e_{1},\dots,e_{n}\rrbracket}$

Here $\alpha^{\natural}$ is the unique graded $X(1)$-algebra homomorphism that
satisfies $\alpha^{\natural}(e_{i})=d^{n}(e_{p_{i}\alpha})$ for $i=1,2$.
Indeed, both composites have the same effect on the regular sequence
$(e_{1},e_{2})$ that generates the augmentation ideal of $(t^{-1}X)(C\times
C)$. So they coincide by Proposition 4.4 (ii). The commutativity of the
diagram then yields

$\alpha^{\natural}(F(e_{1},e_{2}))\ =\ \alpha^{\natural}(d^{2}(e_{\mu}))\ =\
d^{n}(\alpha^{*}(e_{\mu}))\ .$

We derive the four relations of a 2-torsion formal group law by specializing
for five different choices of $\alpha$. When $\alpha$ is the inclusion of the
first factor $i_{1}:C\longrightarrow C\times C$, then
$i_{1}^{\natural}:X(1)\llbracket e_{1},e_{2}\rrbracket\longrightarrow
X(1)\llbracket e_{1}\rrbracket$ sets $e_{2}=0$, and
$i_{1}^{*}(e_{\mu})=e_{1}$, so we get

$F(e_{1},0)\ =\ i_{1}^{\natural}(F(e_{1},e_{2}))\ =\ \ d^{1}(e_{1})\ =\ e_{1}\
.$

When $\alpha$ is the diagonal $\Delta:C\longrightarrow C\times C$, then
$\delta^{\natural}:X(1)\llbracket e_{1},e_{2}\rrbracket\longrightarrow
X(1)\llbracket e_{1}\rrbracket$ sends both $e_{1}$ and $e_{2}$ to $e_{1}$, and
$\delta^{*}(e_{\mu})=0$, so we obtain

$F(e_{1},e_{1})\ =\ \Delta^{\natural}(F(e_{1},e_{2}))\ =\ \ d^{1}(0)\ =\ 0\ .$

When $\alpha$ is the involution $\tau:C\times C\longrightarrow C\times C$ that
interchanges the two factors, then $\tau^{\natural}:X(1)\llbracket
e_{1},e_{2}\rrbracket\longrightarrow X(1)\llbracket e_{1},e_{2}\rrbracket$
interchanges $e_{1}$ and $e_{2}$, and $\tau(e_{\mu})=e_{\mu}$; so we get the
commutativity relation

$F(e_{2},e_{1})\ =\ \tau^{\natural}(F(e_{1},e_{2}))\ =\ \ d^{2}(e_{\mu})\ =\
F(e_{1},e_{2})\ .$

For $\alpha=\mu\times C:C^{3}\longrightarrow C^{2}$, we have $p_{1}(\mu\times
C)=\mu\circ p_{1,2}$, for $p_{1,2}:C^{3}\longrightarrow C^{2}$ the projection
to the first two coordinates, and $p_{2}(\mu\times C)=p_{3}$. So the
homomorphism $(\mu\times C)^{\natural}:X(1)\llbracket
e_{1},e_{2}\rrbracket\longrightarrow X(1)\llbracket
e_{1},e_{2},e_{3}\rrbracket$ sends $e_{1}$ to
$d^{3}(p_{1,2}^{*}(e_{\mu}))=d^{2}(e_{\mu})=F(e_{1},e_{2})$, and it sends
$e_{2}$ to $e_{3}$. So we obtain

$F(F(e_{1},e_{2}),e_{3})\ =\ (\mu\times C)^{\natural}(F(e_{1},e_{2}))\ =\ \
d^{3}((\mu\times C)^{*}(e_{\mu}))\ .$

The analogous argument for $\alpha=C\times\mu$ yields

$F(e_{1},F(e_{2},e_{3}))\ =\ d^{3}((C\times\mu)^{*}(e_{\mu}))\ .$

Because $(\mu\times C)^{*}(e_{\mu})=(C\times\mu)^{*}(e_{\mu})$ (the
multiplication of $C$ is associative), this establishes the associativity
relation $F(F(e_{1},e_{2}),e_{3})=F(e_{1},F(e_{2},e_{3}))$. ∎

######  Example 4.8 (Additive orientations yield additive formal group laws).

Let $(X,a,t)$ be an additive ${\operatorname{el}^{RO}_{2}}$-algebra in the
sense of Definition 3.3, i.e.,

$a_{1}t_{\mu}t_{2}\ +\ t_{1}a_{\mu}t_{2}\ +\ t_{1}t_{\mu}a_{2}\ =\ 0$

in $X(C\times C,2-(p_{1}\oplus\mu\oplus p_{2}))$. Then

$e_{1}+e_{\mu}+e_{2}\ =\
(a_{1}t_{\mu}t_{2}+t_{1}a_{\mu}t_{2}+t_{1}t_{\mu}a_{2})/t_{1}t_{\mu}t_{2}\ =\
0\ .$

So

$F(e_{1},e_{2})\ =\ d^{2}(e_{\mu})\ =\ d^{2}(e_{1}+e_{2})\ =\ e_{1}+e_{2}\ ,$

i.e., the formal group law is the additive formal group law.

##  5\. Geometric fixed points

To establish the universal property of equivariant bordism, we shall make use
of the geometric fixed points of an orientable
${\operatorname{el}^{RO}_{2}}$-algebra, obtained by inverting all the
$a$-classes of all nontrivial characters. For orientable
${\operatorname{el}^{RO}_{2}}$-algebras that arise from global ring spectra,
this construction generalizes the geometric fixed point construction for
equivariant spectra, see Remark 5.3 below, thence the name.

We also discuss various ways in which the power series expansions of the
previous section can be extended to Laurent series expansions on geometric
fixed points. The main result is the integrality criterion of Theorem 5.8 that
allows to detect when a class in the geometric fixed point ring
$\Phi^{A}_{*}X$ arises from a class in $X(A)_{*}$. This criterion will be used
in Theorem 5.14 to show how morphisms of oriented
${\operatorname{el}^{RO}_{2}}$-algebras can be recovered from the induced
morphisms of geometric fixed point rings.

######  Construction 5.1 (Geometric fixed points).

We let $(X,a)$ be an orientable ${\operatorname{el}^{RO}_{2}}$-algebra. For
every elementary abelian 2-group $A$, we define the ring of $A$-geometric
fixed points $\Phi^{A}_{*}X$ as the integer graded part of the localization of
the $I_{A}$-graded ring $X(A,\star)$ obtained by inverting all classes of the
form $a_{\lambda}$ for all nontrivial $A$-characters $\lambda$. Homogeneous
elements in $\Phi^{A}_{k}X$ are thus fractions of the form $x/a_{V}$ for some
$A$-representation $V$ with trivial fixed points and some $x\in X(A,k-V)$.
These elements satisfy the relations

$x/a_{V}\ =\ (x\cdot a_{W})/a_{V\oplus W}\text{\qquad and\qquad}x/a_{V}\cdot
y/a_{W}\ =\ (x\cdot y)/a_{V\oplus W}$

for all $A$-representations $W$ with $W^{A}=0$. In an orientable
${\operatorname{el}^{RO}_{2}}$-algebra, multiplication by the classes
$a_{\lambda}$ is injective; hence also the maps $-/a_{V}\colon
X(A,k-V)\longrightarrow\Phi^{A}_{k}X$ are injective.

The geometric fixed point rings enjoy inflation homomorphisms, i.e., they are
contravariantly functorial for epimorphisms $\beta:B\longrightarrow A$.
Indeed, $\beta^{*}(a_{\lambda})=a_{\lambda\beta}$, so the inflation
homomorphism $\beta^{*}:X(A,\star)\longrightarrow X(B,\star)$ induces a ring
homomorphism $\beta^{*}:\Phi^{A}_{*}X\longrightarrow\Phi^{B}_{*}X$ of
localizations. In terms of the above presentation as fractions, we have

(5.2) $\beta^{*}(x/a_{V})\ =\ \beta^{*}(x)/a_{\beta^{*}(V)}\ .$

We emphasize a crucial difference between $\Phi^{A}_{*}X$, the localization
away from $a$, and the localization $(t^{-1}X)(A)$ away from $t$ discussed in
Construction 4.1: the latter admit restriction homomorphisms for all group
homomorphisms, the former only admit inflations, i.e., restriction along
epimorphisms.

######  Remark 5.3 (Relation to geometric fixed points of equivariant
spectra).

As already mentioned above, the term ‘geometric fixed points’ for the
localization of an orientable ${\operatorname{el}^{RO}_{2}}$-algebra away from
the $a$-classes is motivated by the topological examples. Indeed, for every
compact Lie group $G$ and every genuine $G$-spectrum $R$, a common definition
of the geometric fixed point homotopy groups is as the reduced $G$-equivariant
$R$-homology of a specific based $G$-space $\tilde{E}\mathcal{P}$, see for
example [16, (3.18)]. Here $\tilde{E}\mathcal{P}$ is the unreduced suspension
of a classifying $G$-space for the family of proper closed subgroups of $G$. A
natural homomorphism $R_{*}^{G}\longrightarrow
R_{*}^{G}(\tilde{E}\mathcal{P})$ from the equivariant homotopy groups to the
geometric fixed point homotopy groups is defined as the effect on
$R_{*}^{G}(-)$ of the inclusion of cone points
$S^{0}\longrightarrow\tilde{E}\mathcal{P}$. If $R$ is a ring spectrum in the
homotopy category of genuine $G$-spectra, this map is a morphism of graded
rings, and it factors through an isomorphism from the localization of the
$RO(G)$-graded homotopy ring of $R$ at the pre-Euler classes of all real
$G$-representations $V$ with $V^{G}=0$, see [16, Proposition 3.20].

If $R$ is a global ring spectrum and $\lambda$ a nontrivial character of an
elementary abelian 2-group $A$, then $a_{\lambda}\in
R^{\sharp}(A,-\lambda)=R_{0}^{A}(S^{\lambda})$ is precisely the pre-Euler
class of the $A$-representation $\lambda$, so the geometric fixed point ring
$\Phi^{A}_{*}R^{\sharp}$, in the sense of Construction 5.1, of the orientable
${\operatorname{el}^{RO}_{2}}$-algebra $(R^{\sharp},a)$ is the localization of
the $RO(A)$-graded homotopy ring of $R$ at the pre-Euler classes. Hence the
two geometric fixed point rings $\Phi_{*}^{A}R^{\sharp}$ and
$R_{*}^{A}(\tilde{E}\mathcal{P})$ of an orientable global ring spectrum are
naturally isomorphic.

For a ${\mathbb{Z}}$-graded ring $R$, we now start using the notation

$R(\\!(\theta)\\!)\ =\ R\llbracket\theta\rrbracket[\theta^{-1}]$

for the ${\mathbb{Z}}$-graded Laurent power series in a variable $\theta$ of
degree $-1$, the localization of the power series ring away from the element
$\theta$. So elements of $R(\\!(\theta)\\!)$ of degree $k$ are power series
$\sum_{i\in{\mathbb{Z}}}x_{i}\theta^{i}$ such that $x_{i}\in R_{k+i}$, and
$x_{i}=0$ for all almost all negative values of $i$.

Let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra, and let
$K$ be an elementary abelian 2-group. We will now define a Laurent power
series expansion on the geometric fixed point ring

$d_{K}\ :\ \Phi^{K\times C}_{*}X\ \longrightarrow\
(\Phi^{K}_{*}X)(\\!(\theta)\\!)\ .$

These homomorphisms will be instrumental in detecting effective classes in the
geometric fixed point rings, see Theorem 5.8, and in reconstructing morphisms
of oriented ${\operatorname{el}^{RO}_{2}}$-algebra from their effect on
geometric fixed point rings, see Theorem 5.14.

######  Construction 5.4.

We let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra, and we
let $K$ be an elementary abelian 2-group. The construction of the Laurent
power series expansion requires certain intermediate constructions that we
summarize in the following commutative diagram:

(5.5)
---
$\textstyle{\Psi^{K}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{K}}$$\scriptstyle{i_{1}^{*}}$$\textstyle{(\Psi^{K}X)[e_{2}^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\delta}_{K}}$$\textstyle{\Phi^{K\times
C}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{K}}$$\textstyle{\Phi^{K}X}$$\textstyle{(\Phi^{K}X)\llbracket\theta\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta=0}$$\textstyle{(\Phi^{K}X)(\\!(\theta)\\!)}$

We introduce the graded ring

(5.6) $\Psi^{K}X\ =\ X(K\times C,\star)[t_{2}^{-1},a_{V}^{-1}\colon V^{K}=0]\
,$

the integer graded part of the localization of the $I_{K\times C}$-graded ring
$X(K\times C,\star)$ at all $a$-classes of $(K\times C)$-representations with
trivial $K$-fixed points, and the class $t_{2}=p_{2}^{*}(t)$ of the projection
to the second factor. Inverting the classes $a_{V}$ for all $V^{K}=0$ is the
same as inverting the classes $a_{\nu}$ for all non-zero characters of
$K\times C$ except the projection to the second factor. For $\nu\neq p_{2}$,
the restriction of $\nu$ to $K$ is nontrivial. The restriction homomorphism

$i_{1}^{*}\ :\ X(K\times C,\star)\ \longrightarrow\ X(K,\star)$

along the embedding $i_{1}:K\longrightarrow K\times C$ as the first factor
sends the class $t_{2}$ to 1, and it sends the class $a_{V}$ to the pre-Euler
class of the restriction of $V$ to $K$. So the restriction extends uniquely to
localizations to a morphism of graded rings

$i_{1}^{*}\ :\ \Psi^{K}X\ \longrightarrow\ \Phi^{K}X$

that we denote by the same symbol. The inflation homomorphism

$p_{1}^{*}\ :\ X(K,\star)\ \longrightarrow\ X(K\times C,\star)$

along the projection $p_{1}:K\times C\longrightarrow K$ extends uniquely to a
homomorphism on the localization

$p_{1}^{*}\ :\ \Phi^{K}X\ \longrightarrow\ \Psi^{K}X$

that splits $i_{1}^{*}$. We write $a_{2}=p_{2}^{*}(a)$ for the pre-Euler class
of the projection to the second factor, and we set

$e_{2}\ =\ a_{2}/t_{2}\ \in\ \Psi^{K}_{-1}X\ .$

Because localization is exact, the sequence

$0\ \longrightarrow\ \Psi^{K}_{*+1}X\ \xrightarrow{e_{2}\cdot-}\
\Psi^{K}_{*}X\ \xrightarrow{\ i_{1}^{*}\ }\ \Phi^{K}_{*}X\ \longrightarrow\ 0$

is exact. Hence $e_{2}$ is regular in $\Psi^{K}X$, and the composite of
$p_{1}^{*}:\Phi^{K}_{*}X\longrightarrow\Psi^{K}_{*}X$ and the projection
$\Psi^{K}_{*}X\longrightarrow(\Psi^{K}_{*}X)/(e_{2})$ is an isomorphism. We
let

$\delta_{K}\ :\ \Psi^{K}X\ \longrightarrow\
(\Phi^{K}X)\llbracket\theta\rrbracket$

denote the associated power series expansion in the sense of Definition 4.5.
So $\delta_{K}$ is the unique morphism of graded $\Phi^{K}X$-algebras such
that $\delta_{K}(e_{2})=\theta$. The homomorphism extends uniquely to the
localizations to a homomorphism

$\bar{\delta}_{K}\ :\ (\Psi^{K}X)[e_{2}^{-1}]\ \longrightarrow\
(\Phi^{K}X)(\\!(\theta)\\!)\ .$

The ring $(\Psi^{K}X)[e_{2}^{-1}]$ is the localization of $X(K\times C,\star)$
obtained by inverting all $a$-classes and the class $t_{2}$. Since
$\Phi^{K\times C}X$ is a less drastic localization – formed by inverting all
$a$-classes – it affords a localization morphism $\Phi^{K\times
C}X\longrightarrow(\Psi^{K}X)[e_{2}^{-1}]$ that is localization away from
$t_{2}/a_{2}$. The composite with $\bar{\delta}_{K}$ is the homomorphism

$d_{K}\ :\ \Phi^{K\times C}X\ \longrightarrow\ (\Phi^{K}X)(\\!(\theta)\\!)\ ;$

it is then given by

$d_{K}(x/(a_{V}a_{2}^{n}))\ =\ \delta_{K}(x/(a_{V}t_{2}^{n}))\cdot\theta^{-n}\
,$

for $x\in X(K\times C,k-(V\oplus np_{2}))$ with $V^{K}=0$. We have now
introduced all ring homomorphisms in the diagram (5.5).

We make the following definition:

######  Definition 5.7.

Let $X$ be an orientable ${\operatorname{el}^{RO}_{2}}$-algebra and $A$ an
elementary abelian 2-group. A class in $\Phi_{k}^{A}X$ is effective if it lies
in the image of the homomorphism $-/1:X(A)_{k}\longrightarrow\Phi_{k}^{A}X$.

For every orientable ${\operatorname{el}^{RO}_{2}}$-algebra, the map
$-/1:X(A)_{k}\longrightarrow\Phi_{k}^{A}X$ is injective. So if a class is
effective, then its preimage in the integer graded subring is unique. Said
differently, the maps $-/1:X(A)_{\ast}\longrightarrow\Phi_{*}^{A}X$ form an
isomorphism from the ${\mathbb{Z}}$-graded part of $X(A,\star)$ to the subring
of effective classes.

The next theorem establishes an effectivity criterion for geometric fixed
point classes in terms of the homomorphisms $d_{K}:\Phi^{K\times
C}X\longrightarrow(\Phi^{K}X)(\\!(\theta)\\!)$. Effective classes in
$\Phi^{K\times C}X$ in particular also lie in the ring $\Psi^{K}X$, and hence
their Laurent power series expansion is contained in the subring
$(\Phi^{K}X)\llbracket\theta\rrbracket$, compare the diagram (5.5). Moreover,
$d_{K}$ is designed so that for $u\in X(K\times C)_{*}$, the constant
coefficient is given by

$d_{K}(u/1)(0)\ =\ i_{1}^{*}(u/1)\ =\ i_{1}^{*}(u)/1\ .$

######  Theorem 5.8.

Let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra, and let
$A$ be an elementary abelian 2-group. For every $y\in\Phi^{A}X$, the following
two conditions are equivalent.

  1. (a)

The class $y$ is effective.

  2. (b)

For every isomorphism $\alpha:K\times C\cong A$, the class
$d_{K}(\alpha^{*}(y))$ in $(\Phi^{K}X)(\\!(\theta)\\!)$ is integral, i.e.,
contained in the subring $(\Phi^{K}X)\llbracket\theta\rrbracket$.

###### Proof.

(a)$\Longrightarrow$(b) For every $u\in X(A)_{k}$, the class
$\alpha^{*}(u/1)=\alpha^{*}(u)/1$ lies in $\Psi^{K}X$, so
$d_{K}(\alpha^{*}(u/1))$ lies in $(\Phi^{K}X)\llbracket\theta\rrbracket$,
compare the commutative diagram (5.5).

(b)$\Longrightarrow$(a) We write $y=x/a_{V}$ for some $A$-representation $V$
with $V^{A}=0$, some $k\in{\mathbb{Z}}$, and some $x\in X(A,k-V)$. We let
$\lambda$ be a nontrivial $A$-character, with kernel $K$. We choose an
isomorphism $\alpha:K\times C\cong A$ such that $\lambda\circ\alpha=p_{2}$. We
decompose $\alpha^{*}(V)=U\oplus np_{2}$ for some $(K\times C)$-representation
$U$ with $U^{K}=0$, and some $n\geq 0$. Then

$d_{K}(\alpha^{*}(y))\ =\ d_{K}(\alpha^{*}(x)/a_{\alpha^{*}(V)})\ =\
d_{K}(\alpha^{*}(x)/a_{U}a_{2}^{n})\ =\
\delta_{K}(\alpha^{*}(x)/a_{U}t_{2}^{n})\cdot\theta^{-n}\ .$

This class is integral by hypothesis, and
$\delta_{K}(\alpha^{*}(x)/a_{U}t_{2}^{n})$ is integral by definition. So we
conclude that

$\delta_{K}(\alpha^{*}(x)/a_{U}t_{2}^{n})\ \in\
\theta^{n}\cdot(\Phi^{K}X)\llbracket\theta\rrbracket\ .$

In other words, the first $n$ coefficients of the power series
$\delta_{K}(\alpha^{*}(x)/a_{U}t_{2}^{n})$ vanish. Hence the class
$\alpha^{*}(x)/a_{U}t_{2}^{n}$ is divisible by $e_{2}^{n}$ in the ring
$\Psi^{K}X$.

Since all the $a$-classes are non zero-divisors, we deduce that for some
$m\geq 0$ the element $\alpha^{*}(x)\cdot t_{2}^{m}$ is divisible by
$a_{2}^{n}$ in $X(K\times C,\star)$. Thus the element
$x\cdot(\alpha^{-1})^{*}(t_{2}^{m})=x\cdot t_{\lambda}^{m}$ is divisible by
$(\alpha^{-1})^{*}(a_{2}^{n})=a_{\lambda}^{n}$ in $X(A,\star)$. Because
$\operatorname{res}^{A}_{K}(t_{\lambda}^{m})=1$, Proposition 1.6 (i) shows
that already $x$ is divisible by $a_{\lambda}^{n}$. Since $n$ is the
multiplicity of $p_{2}$ in $\alpha^{*}(V)$, it is also the multiplicity of
$\lambda=(\alpha^{-1})^{*}(p_{2})$ in $V$. We have thus shown that for every
nontrivial $A$-character $\lambda$, the class $x$ is divisible by
$a_{V_{\lambda}}$. Proposition 1.6 (ii) then shows that $x$ is divisible by
$a_{V}$, say $x=z\cdot a_{V}$. Hence $y=x/a_{V}{}=z/1$ is effective. ∎

######  Remark 5.9 (Relation to Boardman’s work in equivariant bordism).

For the two-element group $C$, the statement of Theorem 5.8 simplifies: an
element $y\in\Phi^{C}X$ is effective if and only if the Laurent power series
$d_{1}(y)$ is integral. For the oriented
${\operatorname{el}^{RO}_{2}}$-algebra of equivariant bordism, Boardman
defines a ‘stable bordism $J$-homomorphism’
$J:\Phi_{*}^{C}\mathcal{N}\longrightarrow\mathcal{N}_{*}(\\!(\theta)\\!)$ in
[7, §6], using different notation and geometric arguments. In this
incarnation, the special case $A=C$ of Theorem 5.8 was proved by Boardman in
[7, Theorem 11]. In this sense, our Theorem 5.8 is a vast generalization of
Boardman’s result to elementary abelian 2-groups of arbitrary rank and general
oriented ${\operatorname{el}^{RO}_{2}}$-algebras.

For every oriented ${\operatorname{el}^{RO}_{2}}$-algebra, the class
$t_{\mu}/a_{\mu}\in\Phi^{C\times C}X$ also lies in the ring $\Psi^{C}X$, so
that $d_{C}(t_{\mu}/a_{\mu})=\delta_{C}(t_{\mu}/a_{\mu})$, and this class is a
power series (as opposed to a Laurent power series) in $\theta$ with
coefficients in $\Phi^{C}X$.

######  Definition 5.10.

Let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra. For
$n\geq 0$, we define the classes $\beta_{n}\in\Phi^{C}_{n+1}X$ by

$d_{C}(t_{\mu}/a_{\mu})\ =\ \sum_{n\geq 0}\beta_{n}\cdot\theta^{n}\ .$

For example, $\beta_{0}=i_{1}^{*}(t_{\mu}/a_{\mu})=t/a$. The next theorem
might seem unmotivated right now, but it will enter crucially in the proof of
our main result, Theorem 7.14. In part (ii), what matters is not the precise
formula, but the fact that the classes $\delta_{C}(\mu^{*}(\beta_{n}))$ can be
described purely in terms of the $2$-torsion formal group law and the classes
$\beta_{i}$ themselves.

######  Theorem 5.11.

Let $(X,a,t)$ be an oriented ${\operatorname{el}^{RO}_{2}}$-algebra with
2-torsion formal group law $F$.

  1. (i)

The images of the classes $\beta_{n}$ under the power series expansion
$d_{1}:\Phi^{C}X\longrightarrow X(1)(\\!(\theta)\\!)$ satisfy the relation

$F(\theta,\xi)\cdot\sum_{n\geq 0}\,d_{1}(\beta_{n})\cdot\xi^{n}\ =\ 1$

in the ring $X(1)(\\!(\theta)\\!)\llbracket\xi\rrbracket$.

  2. (ii)

The classes
$\delta_{C}(\mu^{*}(\beta_{n}))\in(\Phi^{C}X)\llbracket\theta\rrbracket$
satisfy the relation

$\sum_{n\geq 0}\delta_{C}(\mu^{*}(\beta_{n}))\cdot\xi^{n}\ =\ \sum_{n\geq
0}\beta_{n}\cdot F(\theta,\xi)^{n}$

in the ring $(\Phi^{C}X)\llbracket\theta,\xi\rrbracket$.

###### Proof.

(i) We specialize Construction 5.4 to $K=\\{1\\}$. Then
$\Psi^{1}X=X(C,\star)[t^{-1}]=(t^{-1}X)(C)$, with the potentially confusing
caveat that the class that is generically called $e_{2}$ in $\Phi^{1}K$ in
Construction 5.4 is called $e_{1}$ in $(t^{-1}X)(C)$. Under these
identifications, the homomorphism $\delta_{1}:\Psi^{1}X\longrightarrow
X(1)\llbracket\theta\rrbracket$ coincides with
$d^{1}:(t^{-1}X)(C)\longrightarrow X(1)\llbracket e_{1}\rrbracket$ up to
renaming of the variables. So for all $x\in X(C,*-k\sigma)$, the relation

(5.12) $d^{1}(x/t^{k})\cdot e_{1}^{-k}\ =\ \delta_{1}(x/t^{k})\langle
e_{1}\rangle\cdot e_{1}^{-k}\ =\ (\delta_{1}(x/t^{k})\cdot\theta^{-k})\langle
e_{1}\rangle\ =\ d_{1}(x/a^{k})\langle e_{1}\rangle$

holds in $X(1)(\\!(e_{1})\\!)$. Here, and in what follows, we write $\langle
e_{1}\rangle$ to indicate a renaming of the variable from $\theta$ to $e_{1}$.

The classes $d^{2}(e_{1})=e_{1}$ and $d^{2}(e_{\mu})=F(e_{1},e_{2})$ both
become invertible in the ring $X(1)(\\!(e_{1})\\!)\llbracket e_{2}\rrbracket$,
so the power series expansion $d^{2}:(t^{-1}X)(C^{2})\longrightarrow
X(1)\llbracket e_{1},e_{2}\rrbracket$ from Construction 4.6 extends uniquely
to a morphism of graded $X(1)$-algebras

$d^{2}\ :\ (t^{-1}X)(C^{2})[e_{1}^{-1},e_{\mu}^{-1}]\ \longrightarrow\
X(1)(\\!(e_{1})\\!)\llbracket e_{2}\rrbracket\ ,$

for which we use the same name.

The ring $(t^{-1}X)(C^{2})[e_{1}^{-1},e_{\mu}^{-1}]$ is the localization of
$X(C^{2},\star)$ obtained by inverting all $t$-classes and the classes $a_{1}$
and $a_{\mu}$. Since $\Psi^{C}X$ is a less drastic localization – formed by
inverting $t_{2}$, $a_{1}$ and $a_{\mu}$ – it affords a localization morphism
$\ell:\Psi^{C}X\longrightarrow(t^{-1}X)(C^{2})[e_{1}^{-1},e_{\mu}^{-1}]$ that
is localization away from $t_{1}/a_{1}$ and $t_{\mu}/a_{\mu}$. This morphism
is explicitly given by

$\ell(x/(a_{1}^{k}a_{\mu}^{l}t_{2}^{m}))\ =\
(x/(t_{1}^{k}t_{\mu}^{l}t_{2}^{m}))\cdot e_{1}^{-k}e_{\mu}^{-l}\ .$

We claim that the following diagram of graded rings commutes:

$\textstyle{\Psi^{C}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell}$$\scriptstyle{\delta_{C}}$$\textstyle{(t^{-1}X)(C^{2})[e_{1}^{-1},e_{\mu}^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{2}}$$\textstyle{(\Phi^{C}X)\llbracket\theta\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)\langle
e_{2}\rangle}$$\scriptstyle{\cong}$$\textstyle{(\Phi^{C}X)\llbracket
e_{2}\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}\llbracket
e_{2}\rrbracket}$$\textstyle{X(1)(\\!(e_{1})\\!)\llbracket e_{2}\rrbracket}$

Indeed, on the one hand, the rectangle commutes after precomposition with
$p_{1}^{*}:\Phi^{C}X\longrightarrow\Psi^{C}X$: the two composites send
$p_{1}^{*}(x/a^{k})$, for $x/a^{k}\in\Phi^{C}X$, to $d_{1}(x/a^{k})\langle
e_{1}\rangle$ and $d^{1}(x/t^{k})\cdot e_{1}^{-k}$, respectively, which agree
by (5.12). On the other hand, both composites in the rectangle send the
regular generator $e_{2}$ of the kernel of
$i_{1}^{*}:\Psi^{C}X\longrightarrow\Phi^{C}X$ to the class with the same name.
So the rectangle commutes by Proposition 4.4 (ii). Now we can prove the claim:

$\displaystyle F(e_{1},e_{2})\cdot\sum_{n\geq 0}\,d_{1}(\beta_{n})(e_{1})\cdot
e_{2}^{n}\ $ $\displaystyle=\ d^{2}(e_{\mu})\cdot d_{1}\llbracket
e_{2}\rrbracket(\delta_{C}(t_{\mu}/a_{\mu})\langle e_{2}\rangle)$
$\displaystyle=\ d^{2}(e_{\mu})\cdot d^{2}(\ell(t_{\mu}/a_{\mu}))\ =\
d^{2}(e_{\mu}\cdot e_{\mu}^{-1})\ =\ 1$

This is the desired relation, up to renaming $e_{1}$ and $e_{2}$ to $\theta$
and $\xi$.

(ii) We consider the ${\mathbb{Z}}$-graded ring

$\Xi(X)\ =\
X(C^{3},\star)[a_{1}^{-1},a_{12}^{-1},a_{\bar{\mu}}^{-1},t_{2}^{-1},t_{23}^{-1},t_{3}^{-1}]\
,$

the integer graded subring of the localization of the $I_{C^{3}}$-graded ring
$X(C^{3},\star)$ by inverting

  * •

the $a$-classes for the projection to the first factor, the sum of the first
two factors, and the total multiplication $\bar{\mu}:C^{3}\longrightarrow C$,

  * •

and the $t$-classes for the sum of the last two factors and the projections to
the second and to the third factor.

The restriction $i_{12}^{*}:X(C^{3},\star)\longrightarrow X(C^{2},\star)$ to
the first two factors satisfies

$i_{12}^{*}(a_{1})\ =\ a_{1}\ ,\quad i_{12}^{*}(a_{12})\ =\
i_{12}^{*}(a_{\bar{\mu}})\ =\ a_{\mu}\ ,\quad i_{12}^{*}(t_{2})\ =\
i_{12}^{*}(t_{23})\ =\ t_{2}\text{\qquad and\qquad}i_{12}^{*}(t_{3})\ =\ 1\ ;$

so it descends to a homomorphism of localizations

$i_{12}^{*}\ :\ \Xi(X)\ \longrightarrow\ \Psi^{C}X\ ,$

where $\Psi^{C}X$ was defined in (5.6). The morphisms

$p_{12}^{*}\ ,\ (C\times\mu)^{*}\ ,\ (\mu\times C)^{*}\ :\ X(C^{2},\star)\
\longrightarrow\ X(C^{3},\star)$

satisfy

$\displaystyle p_{12}^{*}(a_{1})\ $ $\displaystyle=\ (C\times\mu)^{*}(a_{1})\
=\ a_{1}\ ,\quad p_{12}^{*}(a_{\mu})\ =\ (\mu\times C)^{*}(a_{1})\ =\ a_{12}\
,\quad p_{12}^{*}(t_{2})\ =\ t_{2}$ $\displaystyle(C\times\mu)^{*}$
$\displaystyle(a_{\mu})\ =\ a_{\bar{\mu}}\text{\qquad
and\qquad}(C\times\mu)^{*}(t_{2})\ =\ t_{23}$ $\displaystyle(\mu\times C)^{*}$
$\displaystyle(a_{\mu})\ =\ a_{\bar{\mu}}\text{\qquad and\qquad}(\mu\times
C)^{*}(t_{2})\ =\ t_{3}\ .$

So all three extend to the localizations and yield homomorphisms

$p_{12}^{*}\ ,\ (C\times\mu)^{*}\ ,\ (\mu\times C)^{*}\ :\ \Psi^{C}X\
\longrightarrow\ \Xi(X)\ .$

Because $(C\times\mu)\circ i_{12}=p_{12}\circ i_{12}=\operatorname{Id}$, the
first two of these homomorphisms are sections to $i_{12}^{*}$. Because
localization is exact, the class $e_{3}=a_{3}/t_{3}$ is a regular element of
$\Xi(X)$ that generates the kernel of $i_{12}^{*}$. We let

$\partial\ :\ \Xi(X)\ \longrightarrow\ (\Psi^{C}X)\llbracket\xi\rrbracket$

be the power series expansion, in the sense of Definition 4.5, of $\Xi(X)$
with respect to the class $e_{3}$ and the section $p_{12}^{*}$. We will show
that the three subdiagrams of the following large diagram of graded rings
commute:

---
$\textstyle{\Psi^{C}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{C}}$$\scriptstyle{(C\times\mu)^{*}}$$\textstyle{(\Phi^{C}X)\llbracket\theta\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{*}}$$\textstyle{\Phi^{C}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{*}}$$\scriptstyle{\mu^{*}}$$\textstyle{\Xi(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{(\Psi^{C}X)\llbracket\xi\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{C}\llbracket\xi\rrbracket}$$\textstyle{(\Phi^{C}X)\llbracket\theta,\xi\rrbracket}$$\textstyle{\Psi^{C}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mu\times
C)^{*}}$$\scriptstyle{\delta_{C}}$$\textstyle{(\Phi^{C}X)\llbracket\theta\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\langle\xi\rangle}$$\textstyle{(\Phi^{C}X)\llbracket\xi\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{*}\llbracket\xi\rrbracket}$

Here $F_{*}$ is the graded $\Phi^{C}X$-algebra map that sends $\theta$ to the
formal group law $F(\theta,\xi)$. And we continue to use pointy brackets
$\langle-\rangle$ to indicate isomorphisms that rename variables. The left
subdiagram with target $\Xi(X)$ commutes by the relation
$\mu\circ(C\times\mu)=\mu\circ(\mu\times C)$.

Now we show the commutativity of the upper subdiagram featuring $F_{*}$. We
appeal to the uniqueness statement in Proposition 4.4 (ii). After
precomposition with the homomorphism
$p_{1}^{*}:\Phi^{C}X\longrightarrow\Psi^{C}X$, both composites in the upper
part become the homomorphism $-\cdot
1:\Phi^{C}X\longrightarrow(\Phi^{C}X)\llbracket\theta,\xi\rrbracket$, the
inclusion of the coefficient ring. We claim that both composites in the upper
subdiagram also agree on $e_{2}$. The inflation homomorphism

$p_{23}^{*}\ :\ X(C^{2},\star)\ \longrightarrow\ X(C^{3},\star)$

along the projection to the last two factors satisfies

$p_{23}^{*}(t_{1})\ =\ t_{2}\ ,\quad p_{23}^{*}(t_{\mu})\ =\ t_{23}\ ,\quad
p_{23}^{*}(t_{2})\ =\ t_{3}\ ,$

and thus passes to a homomorphism from the localization away from $t$ of the
source to $\Xi(X)$. The composite

(5.13) $(t^{-1}X)(C^{2})\ \xrightarrow{\ p_{23}^{*}\ }\ \Xi(X)\ \xrightarrow{\
\partial\ }\ (\Psi^{C}X)\llbracket\xi\rrbracket\
\xrightarrow{\delta_{C}\llbracket\xi\rrbracket}\
(\Phi^{C}X)\llbracket\theta,\xi\rrbracket$

is a morphism of $X(1)_{*}$-algebras, and it sends the classes $e_{1}$ and
$e_{2}$ to $\theta$ and $\xi$, respectively. Since the regular sequence
$(e_{1},e_{2})$ generates the augmentation ideal of $(t^{-1}X)(C\times C)$,
the uniqueness result from Proposition 4.4 (ii) shows that the composite
(5.13) agrees with

$(t^{-1}X)(C^{2})\ \xrightarrow{\ d^{2}\ }\ X(1)\llbracket
e_{1},e_{2}\rrbracket\ \xrightarrow[\cong]{\langle\theta,\xi\rangle}\
X(1)\llbracket\theta,\xi\rrbracket\ \xrightarrow{\
p^{*}\llbracket\theta,\xi\rrbracket\ }\
(\Phi^{C}X)\llbracket\theta,\xi\rrbracket\ .$

Here $p^{*}:X(1)\longrightarrow\Phi^{C}X$ is the inflation homomorphism. So
these two homomorphisms also agree on the class $e_{\mu}$. Because

$p_{2}\circ(C\times\mu)\ =\ \mu\circ p_{23}\ :\ C^{3}\ \longrightarrow\ C\ ,$

we have $(C\times\mu)^{*}(e_{2})=p_{23}^{*}(e_{\mu})$ in $\Xi(X)$. Thus

$\displaystyle\delta_{C}\llbracket\xi\rrbracket(\partial((C\times\mu)^{*}(e_{2})))\
$ $\displaystyle=\
\delta_{C}\llbracket\xi\rrbracket(\partial(p_{23}^{*}(e_{\mu})))\ =\
p^{*}\llbracket\theta,\xi\rrbracket(d^{2}(e_{\mu})\langle\theta,\xi\rangle)$
$\displaystyle=\ p^{*}\llbracket\theta,\xi\rrbracket(F(\theta,\xi))\ =\
F_{*}(\theta)\ =\ F_{*}(d_{C}(e_{2}))\ .$

Since $e_{2}$ is a regular element that generates the kernel of
$i_{1}^{*}:\Psi^{C}X\longrightarrow\Phi^{C}X$, the uniqueness result from
Proposition 4.4 (ii) shows that the upper part of the diagram commutes.

It remains to show the commutativity of the lower subdiagram, and one more
time we appeal to Proposition 4.4 (ii). This time both composites with the
homomorphism $p_{1}^{*}:\Phi^{C}X\longrightarrow\Psi^{C}X$ become the
composite

$\Phi^{C}X\ \xrightarrow{\ \mu^{*}\ }\ \Psi^{C}X\ \xrightarrow{-\cdot 1}\
(\Psi^{C}X)\llbracket\xi\rrbracket\ ,$

and both composites send the regular element $e_{2}$ to $\xi$. So the lower
part of the big diagram commutes, too.

We can now prove the desired relation:

$\displaystyle\sum_{n\geq 0}d_{C}(\mu^{*}(\beta_{n}))\cdot\xi^{n}\ $
$\displaystyle=\ (d_{C}\circ\mu^{*})\llbracket\xi\rrbracket(\sum_{n\geq
0}\beta_{n}\cdot\xi^{n})\ =\
(\delta_{C}\circ\mu^{*})\llbracket\xi\rrbracket(\delta_{C}(t_{\mu}/a_{\mu})\langle\xi\rangle)$
$\displaystyle=\ F_{*}(\delta_{C}(t_{\mu}/a_{\mu}))\ =\ F_{*}(\sum_{n\geq
0}\beta_{n}\cdot\theta^{n})\ =\ \sum_{n\geq 0}\beta_{n}\cdot
F(\theta,\xi)^{n}\ .$

The third equality is the commutativity of the big diagram, applied to the
element $t/a$. ∎

We conclude this section with a criterion for when a system of morphisms of
geometric fixed point rings arises from a morphism of oriented
${\operatorname{el}^{RO}_{2}}$-algebras. This will be used in the proof of our
main result, Theorem 7.14, showing that equivariant bordism is an initial
oriented ${\operatorname{el}^{RO}_{2}}$-algebra.

We let $\operatorname{el}_{2}^{\operatorname{epi}}$ denote the category whose
objects are all elementary abelian 2-groups, and whose morphisms are
surjective group homomorphisms. An
$\operatorname{el}_{2}^{\operatorname{epi}}$-algebra is a contravariant
functor from $\operatorname{el}_{2}^{\operatorname{epi}}$ to the category of
commutative ${\mathbb{Z}}$-graded ${\mathbb{F}}_{2}$-algebras. For every
oriented ${\operatorname{el}^{RO}_{2}}$-algebra $X$, the collection of
geometric fixed point rings $\Phi^{A}X$ supports inflation homomorphisms (5.2)
along morphisms in $\operatorname{el}^{\operatorname{epi}}_{2}$, so they
naturally form an $\operatorname{el}_{2}^{\operatorname{epi}}$-algebra
$\Phi^{\bullet}X$. Geometric fixed points are often easier to compute than the
${\operatorname{el}^{RO}_{2}}$-algebras themselves. Therefore the following
theorem will be useful to construct morphisms of orientable
${\operatorname{el}^{RO}_{2}}$-algebras through their geometric fixed points.

######  Theorem 5.14.

Let $(X,a,t)$ and $(Y,\bar{a},\bar{t})$ be oriented
${\operatorname{el}^{RO}_{2}}$-algebras. Let
$f^{\bullet}=\\{f^{A}\\}_{A}:\Phi^{\bullet}X\longrightarrow\Phi^{\bullet}Y$ be
a morphism of $\operatorname{el}_{2}^{\operatorname{epi}}$-algebras. Suppose
that the following two conditions are satisfied:

  * (a)

$f^{C}(t/a)=\bar{t}/\bar{a}$

  * (b)

For every elementary abelian 2-group $K$, the following diagram commutes:

$\textstyle{\Phi^{K\times
C}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{K\times
C}}$$\scriptstyle{d_{K}}$$\textstyle{\Phi^{K\times
C}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{K}}$$\textstyle{(\Phi^{K}X)(\\!(\theta)\\!)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{K}(\\!(\theta)\\!)}$$\textstyle{(\Phi^{K}Y)(\\!(\theta)\\!)}$

Then there exists a morphism $f\colon X\longrightarrow Y$ of oriented
${\operatorname{el}^{RO}_{2}}$-algebras such that $\Phi^{A}f=f^{A}$ for all
elementary abelian $2$-groups $A$.

###### Proof.

We start by observing

$f^{A}(t_{\lambda}/a_{\lambda})\ =\ f^{A}(\lambda^{*}(t/a))\ =_{\textrm{(b)}}\
\lambda^{*}(f^{C}(t/a))\ =_{\textrm{(a)}}\ \lambda^{*}(\bar{t}/\bar{a})\ =\
\bar{t}_{\lambda}/\bar{a}_{\lambda}$

for all nontrivial characters $\lambda$ of an elementary abelian 2-group $A$.

We recall from Definition 5.7 that a class $y\in\Phi_{k}^{A}X$ is effective if
it lies in the image of the homomorphism
$-/1:X(A)_{k}\longrightarrow\Phi_{k}^{A}X$, and that the maps
$-/1:X(A)_{*}\longrightarrow\Phi_{*}^{A}X$ form an isomorphism from the
${\mathbb{Z}}$-graded part of $X(A)$ to the subring of effective classes.

Claim 1: for every elementary abelian 2-group $A$, the morphism
$f^{A}:\Phi^{A}X\longrightarrow\Phi^{A}Y$ takes effective classes to effective
classes.

To prove this claim we let $y\in\Phi^{A}X$ be an effective class, and we let
$\alpha:K\times C\cong A$ be an isomorphism. Because the $f$-maps commute with
inflation along $\alpha$ , we have

$d_{K}(\alpha^{*}(f^{A}(y)))\ =\ d_{K}(f^{K\times C}(\alpha^{*}(y)))\
=_{\textrm{(b)}}\ f^{K}(\\!(\theta)\\!)(d_{K}(\alpha^{*}(y)))\ .$

By Theorem 5.8, the class $d_{K}(\alpha^{*}(y))$ in
$(\Phi^{K}Y)(\\!(\theta)\\!)$ is integral. Since
$f^{K}(\\!(\theta)\\!):(\Phi^{K}X)(\\!(\theta)\\!)\longrightarrow(\Phi^{K}Y)(\\!(\theta)\\!)$
preserves integrality, the class $d_{K}(\alpha^{*}(f^{A}(y)))$ in
$(\Phi^{K}Y)(\\!(\theta)\\!)$ is integral. Again by Theorem 5.8, the class
$f^{A}(y)$ is effective.

Because the ring homomorphism $f^{A}$ takes effective classes to effective
classes, there are unique maps $f(A)\colon X(A)_{*}\longrightarrow Y(A)_{*}$
that make the diagram

$\textstyle{X(A)_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-/1}$$\scriptstyle{f(A)}$$\textstyle{Y(A)_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-/1}$$\textstyle{\Phi_{*}^{A}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{A}}$$\textstyle{\Phi_{*}^{A}Y}$

commute. These maps are homomorphisms of graded rings and commute with
inflations because this is the case for the other three morphisms in the
commutative diagram, and because the vertical maps are injective.

Claim 2: For every monomorphism $i:K\longrightarrow A$ between elementary
abelian 2-groups, the following square commutes:

$\textstyle{X(A)_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f(A)}$$\scriptstyle{i^{*}}$$\textstyle{Y(A)_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\textstyle{X(K)_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f(K)}$$\textstyle{Y(K)_{*}}$

It suffices to show this when $i(K)$ has index 2 in $A$, and then we may
assume without loss of generality that $A=K\times C$ and
$i=i_{1}:K\longrightarrow K\times C$ is the embedding as the first summand.
For $u\in X(K\times C)_{*}$ the relation

$f^{K}(\\!(\theta)\\!)(d_{K}(u/1))\ =_{\textrm{(b)}}\ d_{K}(f^{K\times
C}(u/1))\ =\ d_{K}(f(K\times C)(u)/1)$

holds in the ring $(\Phi^{K}_{*}Y)(\\!(\theta)\\!)$. Since both sides of this
equation are integral classes by Theorem 5.8, the relation already holds in
the ring $(\Phi^{K}_{*}Y)\llbracket\theta\rrbracket$; in particular, the
constant terms of both sides agree, and we obtain

$\displaystyle(f(K)(i_{1}^{*}(u))/1\ $ $\displaystyle=\ f^{K}(i_{1}^{*}(u)/1)\
=\ f^{K}(d_{K}(u/1)(0))$ $\displaystyle=\
f^{K}(\\!(\theta)\\!)(d_{K}(u/1))(0)$ ${}_{\textrm{(b)}}$ $\displaystyle=\
d_{K}(f(K\times C)(u)/1)(0)\ =\ i_{1}^{*}(f(K\times C)(u))/1\ .$

Because the map $-/1$ is injective, this concludes the proof of Claim 2.

We can now define the desired morphism of
${\operatorname{el}^{RO}_{2}}$-algebras $f:X\longrightarrow Y$. We let $W$ be
a representation of an elementary abelian 2-group $A$ with $W^{A}=0$. By
Proposition 1.10, every class $x\in X(A,k-W)$ is a homogeneous polynomial with
coefficients in $X(A)_{*}$ of degree $k-W$ in the classes $a_{\lambda}$ and
$t_{\lambda}$, for all nontrivial $A$-characters $\lambda$. In other words,
there is a homogeneous polynomial $g\in
X(A)_{*}[\alpha_{\lambda},\tau_{\lambda}]$ in variables $\alpha_{\lambda}$ and
$\tau_{\lambda}$ of degree $-\lambda$ and $1-\lambda$, respectively, such that
$x=g(a_{\lambda},t_{\lambda})$, i.e., substituting the variables
$\alpha_{\lambda}$ and $\tau_{\lambda}$ by the classes $a_{\lambda}$ and
$t_{\lambda}$ yields $x$. We write $f(A)_{\flat}(g)\in
Y(A)_{*}[\alpha_{\lambda},\tau_{\lambda}]$ for the effect of applying the
graded ring homomorphism $f(A):X(A)_{*}\longrightarrow Y(A)_{*}$ to the
coefficients of the polynomial $g$. The classes $\bar{a}_{\lambda}$ and
$\bar{t}_{\lambda}$ of the ${\operatorname{el}^{RO}_{2}}$-algebra $Y$ can then
be substituted for the variables in $f(A)_{\flat}(g)$, yielding a class in
$Y(A,k-W)$. There is no reason why the polynomial $g$ that presents the class
$x$ should be unique, so we shall need:

Claim 3: The class $f(A)_{\flat}(g)(\bar{a}_{\lambda},\bar{t}_{\lambda})$ in
$Y(A,k-W)$ is independent of how $x=g(a_{\lambda},t_{\lambda})$ is presented
as a polynomial in the classes $a_{\lambda}$ and $t_{\lambda}$.

For every homogeneous polynomial $h\in
Y(A)_{*}[\alpha_{\lambda},\tau_{\lambda}]$ of the appropriate degree we have

$h(\bar{a}_{\lambda},\bar{t}_{\lambda})/\bar{a}_{W}\ =\
h(1,\bar{t}_{\lambda}/\bar{a}_{\lambda})$

in $\Phi^{A}_{k}Y$, i.e., the class
$h(\bar{a}_{\lambda},\bar{t}_{\lambda})/\bar{a}_{W}$ is obtained by
substituting the variables $\alpha_{\lambda}$ and $\tau_{\lambda}$ by the
multiplicative unit $1$ and by $\bar{t}_{\lambda}/\bar{a}_{\lambda}$,
respectively. For $h=f(A)_{\flat}(g)$, this yields

$\displaystyle
f(A)_{\flat}(g)(\bar{a}_{\lambda},,\bar{t}_{\lambda})/\bar{a}_{W}\ $
$\displaystyle=\ f(A)_{\flat}(g)(1,\bar{t}_{\lambda}/\bar{a}_{\lambda})\ =\
f(A)_{\flat}(g)(1,f^{A}(t_{\lambda}/a_{\lambda}))$ $\displaystyle=\
f^{A}(g(1,t_{\lambda}/a_{\lambda}))\ \
f^{A}(g(a_{\lambda},t_{\lambda})/a_{W})\ =\ f^{A}(x/a_{W})\ .$

This geometric fixed point class is thus independent of how $x$ is presented
as a polynomial in the classes $a_{\lambda}$ and $t_{\lambda}$. Since the map
$-/\bar{a}_{W}:Y(A,k-W)\longrightarrow\Phi^{A}_{k}Y$ is injective, this proves
Claim 3.

With Claim 3 established, we can now define a map
$f(A,k-W):X(A,k-W)\longrightarrow Y(A,k-W)$ by

$f(A,k-W)(x)\ =\ f(A)_{\flat}(g)(\bar{a}_{\lambda},\bar{t}_{\lambda})\ ,$

where $g\in X(A)_{*}[\alpha_{\lambda},\tau_{\lambda}]$ is any homogeneous
polynomial of degree $k-W$ such that $x=g(a_{\lambda},t_{\lambda})$. With the
independence of $g$ at hand, it is then straightforward to shows that these
assignments define an $I_{A}$-graded ring homomorphism. And clearly, the map
$f(A,-\lambda):X(A,-\lambda)\longrightarrow Y(A,-\lambda)$ sends $a_{\lambda}$
to $\bar{a}_{\lambda}$, and the map
$f(A,1-\lambda):X(A,1-\lambda)\longrightarrow Y(A,1-\lambda)$ sends
$t_{\lambda}$ to $\bar{t}_{\lambda}$.

Now we check that for varying $A$, these homomorphisms of $I_{A}$-graded rings
are compatible with restriction along homomorphisms $\beta:B\longrightarrow A$
of elementary abelian 2-groups. We already argued above that the relation
$\beta^{*}\circ f(A,\star)=f(B,\star)\circ\beta^{*}$ holds on the integer
graded subrings. The $I_{A}$-graded ring $X(A,\star)$ is generated by its
integer graded subring and the classes $a_{\lambda}$ and $t_{\lambda}$ for all
nontrivial $A$-characters $\lambda$, by Proposition 1.10 (ii). So it suffices
to show that the ring homomorphisms $\beta^{*}\circ f(A,\star)$ and
$f(B,\star)\circ\beta^{*}$ also agree on the classes $a_{\lambda}$ and
$t_{\lambda}$. We distinguish two cases. If the $B$-character $\lambda\beta$
is nontrivial, then $\beta^{*}(a_{\lambda})=a_{\lambda\beta}$, and we obtain

$\beta^{*}(f(A,-\lambda)(a_{\lambda}))\ =\ \beta^{*}(\bar{a}_{\lambda})\ =\
\bar{a}_{\lambda\beta}\ =\ f(B,-(\lambda\beta))(a_{\lambda\beta})\ =\
f(B,-(\lambda\beta))(\beta^{*}(a_{\lambda}))\ .$

The argument for the $t$-classes is the same, mutatis mutandis. If the
$B$-character $\lambda\beta$ is trivial, we need to argue differently. In this
case $\beta^{*}(a_{\lambda})=0$ in $X$, and $\beta^{*}(\bar{a}_{\lambda})=0$
in $Y$, and the relation
$\beta^{*}(f(A,-\lambda)(a_{\lambda}))=f(B,-1)(\beta^{*}(a_{\lambda}))$ holds
as both sides are zero. Also in this case, $\beta^{*}(t_{\lambda})=1$ in $X$
and $\beta^{*}(\bar{t}_{\lambda})=1$ in $Y$, because $t$ and $\bar{t}$ are
inverse Thom classes. Hence $\beta^{*}(f(A,1-\lambda)(t_{\lambda}))$ and
$f(B,0)(\beta^{*}(t_{\lambda}))$ are both the multiplicative unit 1. This
finishes the proof. ∎

##  6\. Equivariant bordism

In this section we introduce the oriented
${\operatorname{el}^{RO}_{2}}$-algebra $\mathcal{N}$ of equivariant bordism,
and we collect various basic properties. The main result of this paper is the
fact that $\mathcal{N}$ is an initial oriented
${\operatorname{el}^{RO}_{2}}$-algebra, see Theorem 7.14 below.

######  Construction 6.1.

Given a finite group $G$ and a based $G$-space $X$, we write
$\widetilde{\mathcal{N}}^{G}_{k}(X)$ for the $k$-th reduced $G$-equivariant
bordism group of $X$. We refer to [29, §2] or [25, Section 6.2] for a detailed
background on equivariant bordism.

For an elementary abelian 2-group $A$, we write a given grading $m\in I_{A}$
as $m=k-V$ for some integer $k$ and an $A$-representation $V$ with trivial
fixed points. Then we set

$\mathcal{N}(A,m)\ =\ \mathcal{N}(A,k-V)\ =\
\widetilde{\mathcal{N}}_{k}^{A}(S^{V})\ ,$

the reduced $A$-equivariant bordism group of dimension $k$ of the onepoint
compactification $S^{V}$. A key point is that this definition is independent
up to preferred isomorphism of the choice of $V$. Indeed, an isomorphism of
$A$-representations $\psi:V\longrightarrow W$ induces an equivariant
homeomorphism $S^{\psi}:S^{V}\longrightarrow S^{W}$, and hence an isomorphism
of bordism groups

$\widetilde{\mathcal{N}}_{k}^{A}(S^{\psi})\ :\
\widetilde{\mathcal{N}}_{k}^{A}(S^{V})\ \longrightarrow\
\widetilde{\mathcal{N}}_{k}^{A}(S^{W})\ .$

Theorem 2.3 (iii) and the fact that equivariant bordism is represented by an
orientable global ring spectrum (Example 6.2) guarantee that the isomorphism
does not depend on the choice of $\psi$. Equivariant bordism is concentrated
in non-negative integer degrees, so the group $\mathcal{N}(A,k-V)$ is trivial
whenever $k$ is negative.

To define the functoriality we recall the suspension isomorphism in
equivariant bordism. A specific class
$d\in\widetilde{\mathcal{N}}_{1}^{A}(S^{1})$ is represented by the identity of
$S^{1}$. The suspension homomorphism

$-\wedge d\ :\ \widetilde{\mathcal{N}}^{A}_{k}(X)\ \longrightarrow\
\widetilde{\mathcal{N}}^{A}_{k+1}(X\wedge S^{1})$

is given by reduced product with the class $d$; it is an isomorphism, for
example by [25, Proposition 6.2.11].

Now we let $\alpha:(B,\alpha^{*}(m))\longrightarrow(A,m)$ be a morphism in the
category ${\operatorname{el}^{RO}_{2}}$. We choose an $A$-representation $V$
and a $B$-representation $W$ so that $m=k-V$ and
$\alpha^{*}(m)=k-\alpha^{*}(V)=l-W$ for integers $k\geq l$. There is then an
isomorphism of $B$-representations

$\psi\ :\ \alpha^{*}(V)\ \xrightarrow{\cong}\ W\oplus{\mathbb{R}}^{k-l}\ .$

We define

$\alpha^{*}\ :\ \mathcal{N}(A,m)\ \longrightarrow\
\mathcal{N}(B,\alpha^{*}(m))$

as the composite

$\widetilde{\mathcal{N}}_{k}^{A}(S^{V})\ \xrightarrow{\ \alpha^{*}\ }\
\widetilde{\mathcal{N}}_{k}^{B}(S^{\alpha^{*}(V)})\
\xrightarrow[\cong]{\widetilde{\mathcal{N}}_{k}^{B}(S^{\psi})}\
\widetilde{\mathcal{N}}_{k}^{B}(S^{W\oplus{\mathbb{R}}^{k-l}})\
\xleftarrow[\cong]{-\wedge d^{k-l}}\ \widetilde{\mathcal{N}}_{l}^{B}(S^{W})\
.$

The first homomorphism is restriction of the action on manifolds and bordisms
along the homomorphism $\alpha$. The homomorphism decorated $-\wedge d^{k-l}$
is an iterated suspension isomorphism. Again by Theorem 2.3 (iii), the
definition of $\alpha^{*}$ does not depend on the choice of $\psi$; this
independence is also needed in verifying that the assignment is compatible
with composition.

The pre-Euler class

$a\ \in\ \mathcal{N}(C,-\sigma)\ =\
\widetilde{\mathcal{N}}^{C}_{0}(S^{\sigma})$

is represented by the inclusion of the isolated fixed point
$\\{0\\}\longrightarrow S^{\sigma}$. The inverse Thom class

$t\ \in\ \mathcal{N}(C,1-\sigma)\ =\
\widetilde{\mathcal{N}}^{C}_{1}(S^{\sigma})$

is represented by the identity of $S^{\sigma}$, with equivariant smooth
structure through its identification with $S({\mathbb{R}}\oplus\sigma)$ by the
stereographic projection.

######  Example 6.2 (Equivariant bordism versus the global Thom spectrum
$\mathbf{mO}$).

The global Thom spectrum $\mathbf{mO}$ is defined in [25, Example 6.1.24]; the
underlying $G$-equivariant homotopy type of $\mathbf{mO}$ has been considered
before, often in different language and notation, for example in [31, Section
5]. The spectrum $\mathbf{mO}$ is the orthogonal spectrum whose $V$-th term is

$\mathbf{mO}(V)\ =\ Th(Gr_{|V|}(V\oplus{\mathbb{R}}^{\infty}))\ ,$

the Thom space of the tautological vector bundle of $|V|$-dimensional
subspaces of $V\oplus{\mathbb{R}}^{\infty}$. We refer to [25, Example 6.1.24]
for the definition of the structure maps and the $E_{\infty}$-multiplication
on $\mathbf{mO}$.

For every compact Lie group $G$, the equivariant Pontryagin-Thom construction
provides a natural transformation of $G$-equivariant homology theories from
$G$-equivariant bordism to the $G$-homology theory represented by the global
Thom spectrum $\mathbf{mO}$, see [25, Construction 6.2.27]. If the group $G$
is a product of a finite group and a torus, then the equivariant Pontryagin-
Thom construction is an isomorphism. This results goes back to Wasserman [33],
a homotopy theoretic proof for finite groups was given by tom Dieck [31, Satz
5], and a proof in the general case can be found in [25, Theorem 6.2.33]. The
reader should beware that the equivariant Pontryagin-Thom construction is
provably not an isomorphism beyond this class of groups; for $G=SU(2)$, the
homotopy theoretic transfer from the maximal torus normalizer to $SU(2)$
yields a class in $\pi_{0}^{SU(2)}(\mathbf{mO})$ that is not in the image, see
[25, Remark 6.2.34].

In any case, the Thom-Pontryagin construction is in particular an isomorphism

$\Theta^{A}(X)\ :\ \widetilde{\mathcal{N}}_{*}^{A}(X)\ \xrightarrow{\ \cong\
}\ \mathbf{mO}_{*}^{A}(X)$

for every elementary abelian 2-group $A$ and every based $A$-CW-complex $X$.
These isomorphisms are furthermore compatible with products and restriction
along group homomorphisms, compare [25, Theorem 6.2.31]. Specializing to the
case where $X=S^{V}$ is the onepoint compactification of an $A$-representation
shows that the Thom-Pontryagin construction is an isomorphism of
${\operatorname{el}^{RO}_{2}}$-algebras

$\Theta\ :\ \mathcal{N}\ \xrightarrow{\ \cong\ }\ \mathbf{mO}^{\sharp}\ .$

Both in $\mathcal{N}$ and in $\mathbf{mO}^{\sharp}$, the pre-Euler class $a$
is obtained by applying the map of based $C$-spaces $S^{0}\longrightarrow
S^{\sigma}$ to the multiplicative unit in the $C$-equivariant coefficients.
Since the Thom-Pontryagin construction is natural in based equivariant maps,
the pre-Euler classes of $\mathcal{N}$ and $\mathbf{mO}^{\sharp}$ match up.

Several times in this paper we will need the identification of the geometric
fixed points of equivariant bordism in terms of fixed point data. This
technique, like several others in the subject, goes back to the pioneering
work of Conner and Floyd.

######  Construction 6.3 (Geometric fixed points of equivariant bordism).

We recall the description of the geometric fixed point rings of equivariant
bordism in terms of non-equivariant bordism groups. We will need it for
elementary abelian 2-groups $A$, but this fixed point description works in the
generality of compact Lie groups, see for example [25, Proposition 6.2.16].
The result takes the form of an isomorphism of graded rings:

(6.4) $\varphi_{A}\ :\ \Phi^{A}_{n}\mathcal{N}\ \xrightarrow{\ \cong\ }\
{\bigoplus}_{j\geq 0}\ \mathcal{N}_{n-j}(Gr_{j}^{A,\perp})$

Here $Gr_{j}^{A,\perp}$ is the moduli space of $j$-dimensional
$A$-representations with trivial fixed points, and
$\mathcal{N}_{n-j}(Gr_{j}^{A,\perp})$ is its non-equivariant bordism group.
More formally, $Gr_{j}^{A,\perp}$ is the $A$-fixed point space of the
Grassmannian of $j$-planes in the orthogonal complement of the $A$-fixed
subspace of a complete $A$-universe. The multiplication on the target arises
from the maps

$Gr_{i}^{A,\perp}\times Gr_{j}^{A,\perp}\ \longrightarrow\ Gr_{i+j}^{A,\perp}$

that classify the direct sum of $A$-representations with trivial fixed points.

Roughly speaking, the isomorphism (6.4) is defined as follows. Elements of
$\Phi^{A}_{n}\mathcal{N}$ are fractions $x/a_{V}$, where
$x\in\widetilde{\mathcal{N}}_{n}^{A}(S^{V})$ for some $n\geq 0$, and some
$A$-representation $V$ with trivial fixed points. We let $(M,h)$ represent the
class $x$, so that $M$ is an $n$-dimensional smooth closed $A$-manifold, and
$h\colon M\longrightarrow S^{V}$ is a continuous $A$-map; such pairs are taken
up to bordism and modulo the subgroup of pairs where $h$ is constant at the
basepoint. Because the $A$-action is smooth, the fixed set $M^{A}$ is a
disjoint union of regularly embedded smooth submanifolds of $M$ of varying
dimensions. The $A$-space $S^{V}$ has only two fixed points 0 and $\infty$,
and only the fixed points over the non-basepoint 0 will play a role. The
Conner–Floyd map (6.4) sends the bordism class $[M,h]$ to $\sum_{j\geq
0}[M^{(j)},\nu_{j}]$, where $M^{(j)}$ is the union of the $(n-j)$-dimensional
components of $M^{A}\cap h^{-1}(0)$, and where $\nu_{j}:M^{(j)}\longrightarrow
Gr_{j}^{A,\perp}$ classifies the $A$-equivariant normal bundle of $M^{(j)}$
inside $M$. We refer to [25, Proposition 6.2.16] for more details on the
construction and for a proof that the Conner–Floyd map is an isomorphism.

The next theorem collects various key properties of equivariant bordism that
are relevant for this paper.

######  Theorem 6.5.

  1. (i)

The pair $(\mathcal{N},a)$ is an orientable
${\operatorname{el}^{RO}_{2}}$-algebra with a unique inverse Thom class $t$.

  2. (ii)

The 2-torsion formal group law of the oriented
${\operatorname{el}^{RO}_{2}}$-algebra $(\mathcal{N},a,t)$ is an initial
2-torsion formal group law.

  3. (iii)

For every elementary abelian 2-group $A$, the morphism of
$\mathcal{N}_{*}$-algebras

$\bigotimes_{\mathcal{N}_{*}}^{\lambda\in A^{\circ}}\ \Phi^{C}_{*}\mathcal{N}\
\longrightarrow\ \Phi^{A}_{*}\mathcal{N}$

that multiplies the inflation homomorphisms for all nontrivial $A$-characters
is an isomorphism.

###### Proof.

(i) As we detailed in Example 6.2, equivariant bordism is represented by the
orientable global ring spectrum $\mathbf{mO}$. Part (i) is thus a special case
of Theorem 2.5. By the general theory, the set of inverse Thom classes of
$(\mathcal{N},a)$ is a torsor over the bordism group $\mathcal{N}^{C}_{1}$;
since this group is trivial, the inverse Thom class is unique.

(ii) We reduce the claim to Quillen’s theorem [24], saying that the formal
group law associated to the preferred, classical real orientation of the non-
equivariant Thom spectrum $MO$ is initial. To this end we recall Quillen’s
result in a form tailored to our purpose.

We let $BO$ denote the classifying space of the infinite orthogonal group. The
Thom spectrum construction assigns to a continuous map $f:X\longrightarrow BO$
from a CW-complex a Thom spectrum $Th(f)$, making it a functor

$Th\ :\ Ho(Top/BO)\ \longrightarrow\ \mathcal{SH}$

from the homotopy category of spaces over $BO$ to the stable homotopy category
of spectra. The functoriality equips the Thom spectrum with a morphism of
spectra $u(f):Th(f)\longrightarrow Th(\operatorname{Id}_{BO})=MO$ that is a
tautological Thom class in $MO$-cohomology, in the sense that the group of
stable maps $[Th(f),MO]=\mathcal{SH}(Th(f),MO)$ is free of rank 1 as a module,
under the Thom diagonal, over the ring $MO^{0}(X)=[\Sigma^{\infty}_{+}X,MO]$,
with $u(f)$ as generator.

In this language, the preferred real orientation of the spectrum $MO$ arises
as the Euler class of the continuous map
$\beta(\sigma-{\mathbb{R}}):BC\longrightarrow BO$ that classifies the group
homomorphism that sends the generator of the group $C$ to the infinite
diagonal matrix with entries $(-1,1,1,\dots)$. Some readers might prefer to
think of continuous maps to $BO$ as classifying virtual vector bundles of rank
0; our name ‘$\beta(\sigma-{\mathbb{R}})$’ for the above map reflects that it
classifies the virtual vector bundle over $BC$ associated to the virtual
$C$-representation $\sigma-{\mathbb{R}}$. The Thom spectrum of
$\beta(\sigma-{\mathbb{R}})$ is then equivalent to the desuspension of the
reduced suspension spectrum of the space $S^{\sigma}\wedge_{C}EC_{+}$ (which
happens to be homotopy equivalent to $BC$), and its Thom class
$u(\beta(\sigma-{\mathbb{R}}))$ lies in the group

$[Th(\beta(\sigma-{\mathbb{R}})),MO]\ =\
[S^{-1}\wedge\Sigma^{\infty}S^{\sigma}\wedge_{C}EC_{+},MO]\ =\
\widetilde{MO}^{1}(S^{\sigma}\wedge_{C}EC_{+})\ .$

Its Euler class $e_{\tau}\in MO^{1}(BC)$, the restriction of the Thom class
along the zero section $BC_{+}\longrightarrow S^{\sigma}\wedge_{C}EC_{+}$, is
the preferred real orientation of $MO$.

By the theory of real-oriented cohomology theories, the ring $MO^{*}(BC\times
BC)$ is then a power series algebra over $MO^{*}$ in the classes
$p_{1}^{*}(e_{\tau})$ and $p_{2}^{*}(e_{\tau})$, and expanding the class
$(B\mu)^{*}(e_{\tau})$ in $p_{1}^{*}(e_{\tau})$ and $p_{2}^{*}(e_{\tau})$
yields a 2-variable power series with coefficients in $MO^{*}$ that is a
2-torsion formal group law. Quillen’s theorem [24, Theorem 3] says that this
2-torsion formal group law is initial.

Now we use the global Thom ring spectrum $\mathbf{MO}$ defined in [25, Example
6.1.7] that represents stable equivariant bordism. For every elementary
abelian 2-group $A$, the equivariant homology theory represented by the global
ring spectrum $\mathbf{MO}$ is the localization of $\mathbf{mO}^{A}_{*}(-)$ at
the inverse Thom classes, compare [25, Corollary 6.1.35]. The localization
maps $\mathbf{mO}^{A}_{*}(-)\longrightarrow\mathbf{MO}^{A}_{*}(-)$ at all
representation spheres define a morphism of oriented
${\operatorname{el}^{RO}_{2}}$-algebras
$(\mathbf{mO}^{\sharp},a,t)\longrightarrow(\mathbf{MO}^{\sharp},a,t)$. Here we
denote the image of the unique inverse Thom class of $\mathbf{mO}^{\sharp}$ by
the same letter $t\in\mathbf{MO}_{1}^{C}(S^{\sigma})$; we alert the reader
that in the orientable ${\operatorname{el}^{RO}_{2}}$-algebra
$\mathbf{MO}^{\sharp}$, the inverse Thom class is far from unique. The
previous morphism exhibits $\mathbf{MO}^{\sharp}$ as the localization away
from $t$ of $\mathbf{mO}^{\sharp}$. In particular, this morphism is an
isomorphism of underlying graded ${\mathbb{F}}_{2}$-algebras, and so it
necessarily takes the 2-torsion formal group law of $\mathbf{mO}^{\sharp}$ to
the 2-torsion formal group law of $\mathbf{MO}^{\sharp}$. So it suffices to
show that the 2-torsion formal group law of the oriented
${\operatorname{el}^{RO}_{2}}$-algebra $(\mathbf{MO}^{\sharp},a,t)$ is an
initial 2-torsion formal group law.

By Theorem 6.1.7 and Remark 6.1.20 of [25], the class $t$ in
$\mathbf{MO}_{1}^{C}(S^{\sigma})$ is an $RO(C)$-graded unit. The inverse of
$t$ is the ‘shifted Thom class’
$u=\bar{\sigma}_{C,\sigma}\in\mathbf{MO}^{1}_{C}(S^{\sigma})$ discussed in
[25, Remark 6.1.20]. We claim that the bundling homomorphism

(6.6) $\beta\ :\ \widetilde{\mathbf{MO}}^{1}_{C}(S^{\sigma})\ \longrightarrow\
\widetilde{MO}^{1}(S^{\sigma}\wedge_{C}EC_{+})$

takes the $C$-equivariant Thom class $u$ to the non-equivariant Thom class
$u(\beta(\sigma-{\mathbb{R}}))$ discussed above. To prove this we refer to the
$C$-equivariant Thom spectrum formalism, a general reference for which is [23,
Chapter X, §3]. We write $\mathbf{BO}_{C}$ for the classifying $C$-space for
virtual $C$-equivariant vector bundles of rank 0. Much like in the non-
equivariant situation, the equivariant Thom construction turns $C$-equivariant
maps $X\longrightarrow\mathbf{BO}_{C}$ into genuine $C$-spectra endowed with a
morphism to $\mathbf{MO}_{C}$, the underlying $C$-spectrum of the global
spectrum $\mathbf{MO}$. An explicit representative for the class $u$ in
$\mathbf{MO}^{1}_{C}(S^{\sigma})$ is the $C$-equivariant map

$\displaystyle S^{\sigma\oplus\sigma}\ $ $\displaystyle\longrightarrow\
Th(Gr_{2}(\sigma\oplus{\mathbb{R}}\oplus\sigma\oplus{\mathbb{R}}))\ =\
\mathbf{MO}(\sigma\oplus{\mathbb{R}})$ $\displaystyle(x,y)\ $
$\displaystyle\longmapsto\ ((x,0,y,0),\sigma\oplus 0\oplus\sigma\oplus 0)$

So $u:S^{\sigma-1}\longrightarrow\mathbf{MO}_{C}$ is the $C$-equivariant Thom
class of the $C$-equivariant map $*\longrightarrow\mathbf{BO}_{C}$ that
classifies the virtual $C$-representation
$2\sigma-(\sigma\oplus{\mathbb{R}})=\sigma-{\mathbb{R}}$. The unstable and
stable bundling maps take equivariant Thom spectra to non-equivariant Thom
spectra, see [23, X Corollary 6.3]. When applied to the $C$-map
$\sigma-{\mathbb{R}}:*\longrightarrow\mathbf{BO}_{C}$, this identifies the
image of
$u:S^{\sigma-1}=Th(\sigma-{\mathbb{R}})\longrightarrow\mathbf{MO}_{C}$ under
the stable bundling map (6.6) with the non-equivariant stable map

$u(\beta(\sigma-{\mathbb{R}}))\ :\
S^{\sigma-1}\wedge_{C}EC_{+}=Th(\beta(\sigma-{\mathbb{R}}))\ \longrightarrow\
MO$

obtained by thomifying $\beta(\sigma-{\mathbb{R}}):BC\longrightarrow BO$. This
proves the claim that $\beta(u)=u(\beta(\sigma-{\mathbb{R}}))$. Naturality of
the bundling homomorphisms for the $C$-equivariant fixed point inclusion
$a:S^{0}\longrightarrow S^{\sigma}$ then shows that the bundling homomorphism

$\beta\ :\ \mathbf{MO}^{C}_{-1}=\mathbf{MO}^{1}_{C}(S^{0})\ \longrightarrow\
\widetilde{MO}^{1}(S^{0}\wedge_{C}EC_{+})=MO^{1}(BC)$

takes the $C$-equivariant Euler class $e=a/t=a\cdot u$ of the oriented
${\operatorname{el}^{RO}_{2}}$-algebra $(\mathbf{MO}^{\sharp},a,t)$ to the
Euler class $e_{\tau}$ that defines the preferred classical orientation of
$MO$.

Now we can wrap up. The formal group law of $\mathbf{MO}^{\sharp}$ is defined
by expanding the class $\mu^{*}(e)=\mu^{*}(a/t)$ as a power series in
$p_{1}^{*}(e)$ and $p_{2}^{*}(e)$. And the formal group law in Quillen’s work
is the expansion of $\mu^{*}(e_{\tau})$ as a power series in
$p_{1}^{*}(e_{\tau})$ and $p_{2}^{*}(e_{\tau})$. Since the bundling maps are
compatible with restriction along group homomorphisms and take $e$ to
$e_{\tau}$, the 2-torsion formal group law of the oriented