Let $X$ be an $\mathbb{E}_{1}$-space. There is a natural equivalence

$\iota:B(X^{\mathrm{op}})\simeq BX.$

such that, for any anti-involution $\tau$ on $X$, the composition

${\overline{B}\tau:BX}$${B(X^{\mathrm{op}})\overset{\iota}{\simeq}BX}$$\scriptstyle{B\tau}$

is an involution on $BX$.

###### Proof.

For each $k\geq 0$, the map $\mathbb{E}_{1}(k)\to\pi_{0}(\mathbb{E}_{1}(k))$
is an equivalence, and hence there is a natural zig-zag of equivalences of
$\mathbb{E}_{1}$-algebras

${B(\pi_{0}(\mathbb{E}_{1}),\mathbb{E}_{1},X)}$${B(\mathbb{E}_{1},\mathbb{E}_{1},X)}$${X.}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

But the $\mathbb{E}_{1}$-structure on the left hand side factors through the
associative operad $\mathcal{A}ss:=\pi_{0}(\mathbb{E}_{1})$, so for
simplicity, we may assume that $X$ is strictly associative. The equivalence
$\iota$ is then induced on the realisation of the nerve $NX$ by the maps

$X^{q}\times\Delta^{q}\longrightarrow
X^{q}\times\Delta^{q},\quad(x_{1},\dots,x_{q},r)\longmapsto(x_{q},\dots,x_{1},\Phi_{q}(r)),$

where $\Phi_{q}:\Delta^{q}\cong\Delta^{q}$ is the linear homeomorphism induced
by reversing the order of the vertices. It is easy to check that the map
$\overline{B}\tau$ indeed defines an involution on $BX$.∎

###  5.2. From the Weiss–Williams spectrum to spaces of $h$-cobordisms

We now recall Vogell’s model for spaces of $h$-cobordisms (cf. [Vog85, p.
296]). A partition of a manifold $M^{d}$ is a triple $(W,F,V)$, where $W$ is a
codimension zero submanifold of $M\times[-1,1]$, $V$ is the closure of the
complement of $W$ and $F^{d}:=W\cap V$. For technical reasons (see
Construction 5.9 below), we require $F$ to be standard near $\partial
M\times[-1,1]$, and that it intersects it in $\partial M\times\\{0\\}$. Let
$H(M)$ denote the simplicial set a $p$-simplex of which is a (locally trivial
smooth) family of partitions of $M$ parametrised by $\Delta^{p}$ such that $W$
is an $h$-cobordism from $M\times\\{-1\\}\times\Delta^{p}$ to $F$. Set
$H(M):=|H(M)|$ and write $H^{s}(M)\subset H(M)$ for the connected component
containing the trivial partition
$*=(M\times[-1,0],M\times\\{0\\},M\times[0,1])$. There is a canonical
involution $\iota_{H}$ given by turning upside down partitions. Namely

$\iota_{H}:H(M)\longrightarrow
H(M),\quad\rho=(W,F,V)\longrightarrow\rho^{*}:=(V^{*},F^{*},W^{*}),$

where $W^{*}$, $F^{*}$ and $V^{*}$ are respectively the images of $W$, $F$ and
$V$ under the reflection $r=\mathrm{Id}_{M}\times-1$. For the smooth case, we
will also need a small variant of this $h$-cobordism space, denoted
$H_{\mathsf{col}}(M)$, a point of which consists of a partition
$\rho=(W,F,V)\in H(M)$ together with a bicollar of $F$ for $W$ and $V$ which
is standard near $\partial M\times[-1,1]$. The forgetful map
$H_{\mathsf{col}}(M)\to H(M)$ is a weak equivalence by the contractibility of
the space of collars.

Fix an embedding
$M^{d}\subset{\scaleobj{0.97}{\mathbb{R}}}^{N}\subset{\scaleobj{0.97}{\mathbb{R}}}^{\infty}$
and recall that $B\mathrm{Diff}_{\partial}(M)$ admits a model as the moduli
space of manifolds embedded in ${\scaleobj{0.97}{\mathbb{R}}}^{\infty}$ which
are abstractly diffeomorphic to $M$ relative to the boundary $\partial M$.
Similarly
$B\mathrm{Diff}_{\partial}^{b}(M\times{\scaleobj{0.97}{\mathbb{R}}})$ is the
moduli space of manifolds embedded in
${\scaleobj{0.97}{\mathbb{R}}}^{\infty}\times{\scaleobj{0.97}{\mathbb{R}}}$
which are abstractly diffeomorphic to
$M\times{\scaleobj{0.97}{\mathbb{R}}}\subset{\scaleobj{0.97}{\mathbb{R}}}^{\infty}\times{\scaleobj{0.97}{\mathbb{R}}}$
boundedly with respect to the ${\scaleobj{0.97}{\mathbb{R}}}$-direction and
relative to the boundary $\partial M\times{\scaleobj{0.97}{\mathbb{R}}}$ (this
is proved in Appendix A.2). For the remaining of this section, we will denote
by $\underline{{\scaleobj{0.97}{\mathbb{R}}}}$ the bounded direction, i.e.,
the last coordinate in
${\scaleobj{0.97}{\mathbb{R}}}^{\infty}\times{\scaleobj{0.97}{\mathbb{R}}}=:{\scaleobj{0.97}{\mathbb{R}}}^{\infty}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$.
There is a natural map
$-\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}:B\mathrm{Diff}_{\partial}(M)\xhookrightarrow{}B\mathrm{Diff}^{b}_{\partial}(M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}})$
given by sending a manifold $N\subset{\scaleobj{0.97}{\mathbb{R}}}^{\infty}$
to
$N\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}\subset{\scaleobj{0.97}{\mathbb{R}}}^{\infty}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$.

Suppose now we are given some partition $\rho=(W,F,V)\in H^{s}(M)_{0}$ of
$M\times[-1,1]\subset{\scaleobj{0.97}{\mathbb{R}}}^{N}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$.
Then, in particular, as $W$ is an $s$-cobordism from $M$ to $F$ rel boundary,
the manifold
$F\subset{\scaleobj{0.97}{\mathbb{R}}}^{N}\times{\scaleobj{0.97}{\mathbb{R}}}\subset{\scaleobj{0.97}{\mathbb{R}}}^{\infty}$
gives rise to a point in $B\mathrm{Diff}_{\partial}(M)$; more precisely, the
image of the embedding

$i_{\rho}:F\subset M\times
I\subset{\scaleobj{0.97}{\mathbb{R}}}^{N}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}\cong{\scaleobj{0.97}{\mathbb{R}}}^{N+1}\subset{\scaleobj{0.97}{\mathbb{R}}}^{\infty},$

is a point in $B\mathrm{Diff}_{\partial}(M)$, where the isomorphism
${\scaleobj{0.97}{\mathbb{R}}}^{N}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}\cong{\scaleobj{0.97}{\mathbb{R}}}^{N+1}$
identifies $\underline{{\scaleobj{0.97}{\mathbb{R}}}}$ with the last
coordinate in ${\scaleobj{0.97}{\mathbb{R}}}^{N+1}$. We now construct a point
in
$B\mathrm{Diff}_{\partial}^{b}(M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}})$
by ”extending $W$ towards infinity”: Consider the embedding of $F\times[0,1]$
into
${\scaleobj{0.97}{\mathbb{R}}}^{N}\times{\scaleobj{0.97}{\mathbb{R}}}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$
given by

$R:F\times[0,1]\xhookrightarrow{}{\scaleobj{0.97}{\mathbb{R}}}^{N}\times{\scaleobj{0.97}{\mathbb{R}}}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}},\quad(x,t)\longmapsto\underline{e}+(\mathrm{Id}_{{\scaleobj{0.97}{\mathbb{R}}}^{N}}\times
Q_{-\pi t/2})(x-\underline{e}),$

where
$Q_{\theta}:{\scaleobj{0.97}{\mathbb{R}}}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}\cong{\scaleobj{0.97}{\mathbb{R}}}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$
is the rotation matrix $\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}$, and $\underline{e}$ denotes the unit
length vector in $\underline{{\scaleobj{0.97}{\mathbb{R}}}}$. Write
$r:=R\mid_{F\times\\{1\\}}:F\hookrightarrow{\scaleobj{0.97}{\mathbb{R}}}^{N+1}\times\underline{\\{1\\}}$,
and consider

$S:F\times[1,+\infty)\xhookrightarrow{}{\scaleobj{0.97}{\mathbb{R}}}^{N+1}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}},\quad(x,t)\longmapsto\left\\{\begin{array}[]{cc}r(x)+(1-t)\cdot
e_{N+1}+(t-1)\cdot\underline{e},&t\in[1,2],\\\
r(x)-e_{N+1}+(t-1)\cdot\underline{e},&t\geq 2.\end{array}\right.$

Finally consider the region
$D\subset{\scaleobj{0.97}{\mathbb{R}}}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$
given by tuples $(u,v)$ with

$u\geq 0,\quad u\leq 2-v,\quad\text{and if $0\leq u\leq 1$, then
$u\leq(1-(v-1)^{2}))^{1/2}$.}$

Then we define a topological manifold
$\widehat{a}(\rho)\subset{\scaleobj{0.97}{\mathbb{R}}}^{N+1}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$,
depicted in Figure 2, by

$\widehat{a}(\rho):=M\times\\{0\\}\times(-\infty\times-1]\cup W\cup
R(F\times[0,1])\cup S(F\times[1,+\infty))\cup\partial M\times D.$

Figure 2. Depiction of the topological manifold $\widehat{a}(\rho)$ for
$\rho\in H^{s}(M)$.

Now if $\rho$ is a collared partition, i.e. a point in
$H_{\mathsf{col}}^{s}(M)$, one can use the collar of $F$ to smooth out the
corners of the topological manifold $\widehat{a}(\rho)$, and thus obtain a
smooth manifold
$\widetilde{a}(\rho)\subset{\scaleobj{0.97}{\mathbb{R}}}^{N+1}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$
with the same boundary as $M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$,
and which is boundedly diffeomorphic to $M\times{\scaleobj{0.97}{\mathbb{R}}}$
relative to the boundary. This construction can be done simplex-wise in
$H^{s}_{\mathsf{col}}(M)\simeq H^{s}(M)$, and so up to weak equivalence gives
rise to a (simple) Alexander trick-like map

(5.2) $\displaystyle\mathrm{alex}^{s}:H^{s}(M)$
$\displaystyle\overset{\simeq_{0}}{\longrightarrow}\Theta(M)_{0}=\mathrm{Diff}^{b}_{\partial}(M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}})/\mathrm{Diff}_{\partial}(M):=\operatorname{hofib}(B\mathrm{Diff}_{\partial}(M)\to
B\mathrm{Diff}_{\partial}^{b}(M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}})),$
(5.6) $\displaystyle\rho=(W,F,V)$
$\displaystyle\longmapsto\left(i_{\rho}(F),\gamma_{V}:[-\infty,\infty]\ni
t\mapsto\left\\{\begin{array}[]{cc}M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}},&t=-\infty,\\\
\widetilde{a}(\rho)-t\cdot\underline{e},&-\infty<t<+\infty,\\\
i_{\rho}(F)\times\underline{{\scaleobj{0.97}{\mathbb{R}}}},&t=+\infty.\end{array}\right.\right)$

where we can regard the path $\gamma_{V}$ as a $1$-simplex in
$B\mathrm{Diff}_{\partial}^{b}(M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}})$
from $M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$ to
$F\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$. We adopt the convention
that $\pm\infty+r\equiv\pm\infty$ for any real number
$r\in{\scaleobj{0.97}{\mathbb{R}}}$. Upon noting the well-known equivalence
$H^{s}(M)\simeq BC(M)$ (cf. [Vog85, Prop. 2.1]), the map (5.2) is clearly a
delooping of the Alexander trick-like equivalence
$C(M)\simeq\Omega(\mathrm{Diff}_{\partial}^{b}(M\times{\scaleobj{0.97}{\mathbb{R}}})/\mathrm{Diff}(M))$
of [WW88, Prop. 1.10] (i.e. (5.2) is a model for the map $B(\mathrm{alex})$ of
Proposition 3.8 when $V=0$), and therefore it is an equivalence onto the
basepoint component. We keep track of the $s$-cobordism space $H^{s}(M)$ in
the notation of $\mathrm{alex}^{s}$ as we will later extend this map to the
whole of $H(M)$ whenever $M^{d}$ is of the form $M^{\prime}\times I$ with
$d\geq 5$. Let us first prove the following.

######  Lemma 5.6.

The map (5.2) is equivariant up to homotopy for $\iota_{H}$ in the domain and
$\tau_{WW}$ in the target.

###### Proof.

We will give an argument only in the topological setting; in the smooth
setting one works with $H_{\mathsf{col}}(M)$ instead to smooth out corners,
and uses smooth approximations of the continuous functions that will appear in
proof below. We will however state the argument in the smooth setting to
simplify notation. We will also assume at any point in the argument where it
is necessary that a partition $(W,F,V)$ is equipped with some bicollar of $F$
in $W$ and $V$.

The Weiss–Williams involution on
$\mathrm{Diff}^{b}_{\partial}(M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}})/\mathrm{Diff}_{\partial}(M)$
is induced by the identity on $B\mathrm{Diff}_{\partial}(M)$ and the
involution $U\mapsto
U^{*}=(\mathrm{Id}_{{\scaleobj{0.97}{\mathbb{R}}}^{\infty}}\times(-1)_{\underline{{\scaleobj{0.97}{\mathbb{R}}}}})(U)$
on
$B\mathrm{Diff}^{b}_{\partial}(M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}})$.
Then for $\rho=(W,F,V)\in H^{s}(M)$,

$\tau_{WW}\circ\mathrm{alex}^{s}(W,F,V)=(i_{\rho}(F),\gamma_{W}^{*}),\qquad\mathrm{alex}^{s}\circ\iota_{H}(W,F,V)=(i_{\rho^{*}}(F^{*}),\gamma_{V^{*}}),$

where $\gamma_{W}^{*}(t):=(\gamma_{W}(t))^{*}$. We have depicted the paths
$\gamma_{W}^{*}$ and $\gamma_{V^{*}}$ in Figure 3. We need to find a path
$\eta:[-1,1]\to B\mathrm{Diff}_{\partial}(M)$ from $i_{\rho}^{*}(F^{*})$ to
$i_{\rho}(F)$ and a homotopy $\\{H_{s}(-)\\}_{-1\leq s\leq 1}$ from
$\gamma_{V^{*}}(-)$ to $\gamma_{W}^{*}(-)$ such that
$H_{s}(-\infty)=M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$ and
$H_{s}(+\infty)=\eta(s)\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$ for
all $s\in[-1,1]$.

Figure 3. Paths $\gamma_{W}^{*}$ and $\gamma_{V^{*}}$ in
$B\mathrm{Diff}^{b}_{\partial}(M\times\mathbb{R})$. The arrow indicates the
direction of the path as time increases.

For $\eta$, we use the last two coordinates in
${\scaleobj{0.97}{\mathbb{R}}}^{N+2}$ to do a half rotation of that plane.
More explictly,
$\eta(s):=(\mathrm{Id}_{{\scaleobj{0.97}{\mathbb{R}}}^{N}}\times
Q_{\pi\cdot(s+1)/2})(i_{\rho}(F))$ where
$Q_{\theta}:{\scaleobj{0.97}{\mathbb{R}}}^{2}\cong{\scaleobj{0.97}{\mathbb{R}}}^{2}$
is as before. View ${\scaleobj{0.97}{\mathbb{R}}}^{\infty}$ as
${\scaleobj{0.97}{\mathbb{R}}}^{\infty}\times\underline{\\{0\\}}\subset{\scaleobj{0.97}{\mathbb{R}}}^{\infty}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$
and write $N:=\bigcup_{s\in[-1,1]}\eta(s)+s\cdot\underline{e}$. For
$X\subset{\scaleobj{0.97}{\mathbb{R}}}^{\infty}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$,
write $X\mid_{[a,b]}$ for
$X\cap({\scaleobj{0.97}{\mathbb{R}}}^{\infty}\times\underline{[a,b]})$. Then
consider the compact manifold

$U_{\rho}:=\left(\widehat{a}(\rho^{*})\mid_{[-1,2]}-3\cdot\underline{e}\right)\cup
N\cup\left((\widehat{a}(\rho)\mid_{[-1,2]})^{*}+3\cdot\underline{e}\right)$

depicted in Figure 4. We can straighten $U_{\rho}-(V^{*}\cup W^{*})\cong
F^{*}\times(-3,3)$ to a surface of revolution in
${\scaleobj{0.97}{\mathbb{R}}}^{N}\times{\scaleobj{0.97}{\mathbb{R}}}^{2}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$
about a rotation in
${\scaleobj{0.97}{\mathbb{R}}}^{3}\cong{\scaleobj{0.97}{\mathbb{R}}}^{2}\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$.
Using the bicollar of $F^{*}$ in $W^{*}$ and $V^{*}$ to gradually eat up this
surface of revolution, we obtain a canonical path $\psi$ from $U_{\rho}$ to
$M\times\underline{[-4,4]}=M\times\underline{[-4,-1]}\cup V^{*}\cup W^{*}\cup
M\times\underline{[1,4]}$ in the moduli space of manifolds inside
${\scaleobj{0.97}{\mathbb{R}}}^{N+2}\times\underline{[-4,4]}$ which are
diffeomorphic to $M\times\underline{[-4,4]}$ relative to its boundary
$M\times\underline{\\{-4\\}}\cup\partial M\times\underline{[-4,4]}\cup
M\times\underline{\\{4\\}}$.

Figure 4. Depiction of the manifold $U_{\rho}$. Proceed with caution: the part
of the picture corresponding to $N$ takes place in an extra dimension that we
are unable to depict accurately.

We now describe the homotopy $\\{H_{s}(-)\\}_{-1\leq s\leq 1}$. Fix some
homeomorphism $l:[-1,1]\cong[-\infty,+\infty]$, and assume that the path
$\psi$ from $M\times\underline{[-4,4]}$ to $U_{\rho}$ just described is
parametrised by $[-\infty,+\infty]$. Then $H_{s}(-)$ is the concatenation of
two paths $H_{s}^{(1)}(-)$ and $H_{s}^{(2)}(-)$ in
$B\mathrm{Diff}^{b}_{\partial}(M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}})$:
roughly speaking the path $H_{s}^{(1)}(-)$ performs $\psi(-)$ on
$M\times\underline{[-4,4]}+l(s)\cdot\underline{e}\subset
M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$ (if $s=\pm 1$,
$H^{(1)}_{s}(-)$ is constant on
$M\times\underline{{\scaleobj{0.97}{\mathbb{R}}}}$). The path $H_{s}^{(2)}(-)$
starts at $H^{(1)}_{s}(+\infty)$, and sends
$H^{(1)}_{s}(+\infty)\mid_{(-\infty,l(s)+s]}$ and
$H^{(1)}_{s}(+\infty)\mid_{[l(s)+s,+\infty)}$ towards
${\scaleobj{0.97}{\mathbb{R}}}^{N+2}\times\underline{\pm\infty}$,
respectively, extending by $H^{(1)}_{s}(+\infty)\mid_{l(s)+s}$ times an
interval of diverging length. The resulting paths $H_{\pm 1}(-)=H^{(1)}_{\pm
1}(-)\cdot H^{(2)}_{\pm 1}(-)$ are reparametrisations of $\gamma_{V^{*}}$ and
$\gamma_{W^{*}}$ (the reparametrisations only depend on our choice of
homeomorphism $l:[-1,1]\cong[-\infty,+\infty]$ and the parametrisation of the
path $\psi$). Thus $\eta$ and $H$ give rise to the required homotopy
$\tau_{WW}\circ\mathrm{alex}^{s}\simeq\mathrm{alex}^{s}\circ\iota_{H}$. ∎

Recall that $H^{s}(M)\simeq BC(M)$. Now if $P\subset M$ is a codimension zero
embedding and $p\leq d-3$ (in the notation of Theorem A), then
$C\mathrm{Emb}(P,M)\simeq\operatorname{hofib}(H^{s}(\overline{M-P})\to
H^{s}(M))$ by the isotopy extension sequence (3.3), and therefore
$C\mathrm{Emb}(P,M)$ inherits an involution $\iota_{H}$ up to weak
equivalence. The map (5.2) is functorial with respect to codimension zero
embeddings, so the following diagram is commutative:

(5.7)
${H^{s}(\overline{M-P})}$${\Theta(\overline{M-P})_{0}}$${\Omega^{\infty}(\Theta(\overline{M-P}))}$${H^{s}(M)}$${\Theta(M)_{0}}$${\Omega^{\infty}(\Theta(M)).}$$\scriptstyle{\mathrm{alex}^{s}}$$\scriptstyle{\mathrm{alex}^{s}}$

######  Corollary 5.7.

If $\dim P=d$, the vertical homotopy fibre of the horizontal compositions in
(5.7) gives a map

$\mathrm{alex}:C\mathrm{Emb}(P,M)\longrightarrow\Omega^{\infty}(\Sigma^{-1}\Theta(M,M-P))$

which is $\phi_{C\mathrm{Emb}}(d,p)$-connected and $C_{2}$-equivariant up to
homotopy.

###### Proof.

The connectivity of this map is the content of Proposition 3.9. It is
equivariant as both $\mathrm{alex}^{s}:H^{s}(M)\to\Omega^{\infty}(\Theta(M))$
and $\mathrm{alex}^{s}:H^{s}(\overline{M-P})\to\Omega^{\infty}(\Theta(M-P))$
are by Lemma 5.6. ∎

######  Warning 5.8.

There is a canonical involution in the concordance space $C(M)$ given by
turning upside down a concordance and precomposing by the inverse of the top
diffeomorphism (see e.g. [Vog85, p. 296]). The restriction map $C(M)\to
C\mathrm{Emb}(P,M)$ is _not_ $C_{2}$-equivariant with respect to $\iota_{C}$
and $\iota_{H}$—rather, it is anti-equivariant. This may seem to contradict
[Vog85, Prop. 2.2], but what Vogell really proves there is that there is a
homotopy $C_{2}$-equivariant equivalence
$C(M)\approx\Omega^{\sigma}H(M):=\mathrm{Map}_{*}(S^{\sigma},H(M))$, where we
recall $S^{\sigma}$ stands for the representation sphere of the
$1$-dimensional sign representation $\sigma$. This is due to an extra flip in
the loop component that he introduces at the end of the proof of the
proposition.

Let us now extend the map (5.2) to the whole $h$-cobordism space in some
cases. This will not be strictly necessary for the proof of Theorem B, so the
reader may want to skip the rest of this section on first reading.

######  Construction 5.9.

Let $M^{d}$ be a smooth manifold with $d\geq 4$ and $\pi_{1}(M)=:\pi$, and let
$J$ denote a closed interval. The $h$-cobordism space $H(M\times J)$ is an
$\mathbb{E}_{1}$-space with respect to stacking partitions in the
$J$-direction, denoted simply by $+_{J}$. Remember that we imposed the
technical assumption on a partition $\rho=(W,F,V)\in H(M\times J)$ to be such
that the intersection of $F$ with $\partial(M\times J)\times[-1,1]$ is
standard and happens exactly at $\partial(M\times J)\times\\{0\\}$, so this
makes $+_{J}$ well-defined. As $\dim(M\times J)\geq 5$, the
$\mathbb{E}_{1}$-space $(H(M\times J),+_{J})$ is in fact group-like by the
$h$-cobordism theorem, with $\pi_{0}(H(M\times J))\cong\mathrm{Wh}(\pi)$. This
last isomorphism is in fact one of unital monoids by the additivity of
Whitehead torsion (cf. [Coh73, Thm. 23.1]), where the unit in
$\pi_{0}(H(M\times J))$ is the path-component of the standard partition
$*=(M\times J\times[-1,0],M\times J\times\\{0\\},M\times J\times[0,1])$.
Moreover, if $[\rho]\in\mathrm{Wh}(\pi)$ denotes the Whitehead torsion of
$\rho\in H(M\times J)$, i.e. the path-component of $\rho$ in
$\pi_{0}(H(M\times J))$, then

(5.8) $[\iota_{H}(\rho)]=(-1)^{d}\overline{[\rho]},$

where $\overline{(-)}$ is Milnor’s involution on $\mathrm{Wh}(\pi)$ (see
Warning 5.18). This follows by the duality formula of [Mil66, §10] and the
fact that if $\rho=(W,F,V)$, then the Whitehead torsion of the $h$-cobordism
$W$ with respect to $M\times J$ is (roughly) minus that of $V$ with respect to
$F$ (cf. [Mil66, Lem. 7.8]).

On the other hand, the space $\Theta(M\times
J)_{0}:=\mathrm{Diff}_{\partial}^{b}(M\times
J\times{\scaleobj{0.97}{\mathbb{R}}})/\mathrm{Diff}_{\partial}(M\times J)$ is
also an $\mathbb{E}_{1}$-group under $+_{J}$, with $\pi_{0}(\Theta(M\times
J)_{0})\cong\mathrm{Wh}(\pi)$ as groups. Explicitly, the identification is

${\pi_{0}\left(\Omega\frac{\mathrm{Diff}^{b}_{\partial}(M\times{\scaleobj{0.97}{\mathbb{R}}}^{1,1})}{\mathrm{Diff}^{b}_{\partial}(M\times{\scaleobj{0.97}{\mathbb{R}}}^{1,0})}\right)}$${\pi_{0}\left(\frac{\mathrm{Diff}_{\partial}^{b}(M\times
J\times{\scaleobj{0.97}{\mathbb{R}}}^{0,1})}{\mathrm{Diff}_{\partial}(M\times
J)}\right)}$${\pi_{0}(C^{b}(M\times{\scaleobj{0.97}{\mathbb{R}}}^{1,0}))}$${\mathrm{Wh}(\pi).}$${\pi_{0}\left(\Omega^{\sigma}\frac{\mathrm{Diff}^{b}_{\partial}(M\times{\scaleobj{0.97}{\mathbb{R}}}^{1,1})}{\mathrm{Diff}^{b}_{\partial}(M\times{\scaleobj{0.97}{\mathbb{R}}}^{1,0})}\right)}$$\scriptstyle{\cong}$[WW88,
Prop 1.5][WW88, Prop 1.8]$\scriptstyle{\cong}$$\scriptstyle{\cong}$[WW88, Cor.
5.3]

Here
${\scaleobj{0.97}{\mathbb{R}}}^{a,b}={\scaleobj{0.97}{\mathbb{R}}}^{a}\oplus
b\cdot\sigma$ as in Remark 2.1, and $C^{b}(-)$ is the space of bounded
concordances, which is equipped with the concordance involution $\iota_{C}$ of
Warning 5.8. The first and third map are $C_{2}$-equivariant, the second
equality is so only after introducing a minus sign, and the last one after
introducing a $(-1)^{d+1}$ (the homomorphism
$\beta:\pi_{0}(C^{b}(M\times{\scaleobj{0.97}{\mathbb{R}}}^{1,0}))\cong\pi_{0}(H(M\times
I))$ introduced in the proof of [WW88, Cor. 5.3] identifies the concordance
and $h$-cobordism directions, so the claim follows by the duality formula
[Mil66, §10]). All in all, for $\phi\in\Theta(M\times J)_{0}$ we have

(5.9) $[\tau_{WW}(\phi)]=(-1)^{d}\overline{[\phi]}.$

We construct an equivalence

(5.10) $\mathrm{alex}:H(M\times
J)\overset{\sim}{\longrightarrow}\Theta(M\times
J)_{0}=\mathrm{Diff}_{\partial}^{b}(M\times
J\times{\scaleobj{0.97}{\mathbb{R}}})/\mathrm{Diff}_{\partial}(M\times J)$

as follows: for each $\kappa\in\mathrm{Wh}(\pi)$, fix elements $p_{\kappa}\in
H(M\times J)$ and $\phi_{\kappa}\in\mathrm{Diff}_{\partial}^{b}(M\times
J\times{\scaleobj{0.97}{\mathbb{R}}})/\mathrm{Diff}_{\partial}(M\times J)$
with $[p_{\sigma}]=[\phi_{\kappa}]=\kappa$. Fix also paths in $H(M)$ between
$\iota_{H}(p_{\kappa})$ and
$p_{[\iota_{H}(p_{\kappa})]}=p_{(-1)^{d}\overline{\kappa}}$ (by (5.8)), and
paths in $\Theta(M\times J)_{0}$ between $\tau_{WW}(\phi_{\kappa})$ and
$\phi_{[\tau_{WW}(\phi_{\kappa})]}=\phi_{(-1)^{d}\overline{\kappa}}$ (by
(5.9)). When $\kappa=0\in\mathrm{Wh}(\pi)$, we arrange $p_{0}$ and $\phi_{0}$
to be the basepoints in their respective spaces, and the loops around them
that we have just fixed to be constant. Then define

$\mathrm{alex}(\rho):=\mathrm{alex}^{s}(\rho+_{J}p_{-[\rho]})+_{J}\phi_{[\rho]},\qquad\rho\in
H(M\times J).$

######  Lemma 5.10.

The map (5.10) is $C_{2}$-equivariant up to homotopy. Moreover, it is indeed
an equivalence and the composition
$H^{s}(M)\xhookrightarrow{}H(M)\overset{\mathrm{alex}}{\longrightarrow}\Theta(M\times
J)_{0}$ is canonically homotopic to the map $\mathrm{alex}^{s}$ of (5.2).

###### Proof.

For $\phi,\psi\in\Theta(M\times J)_{0}$, let us write $\phi\sim\psi$ to
indicate that there is a path between these two points in that space. The
paths we have fixed above, together with the observation that the involutions
$\iota_{H}$ and $\tau_{WW}$ respect the $\mathbb{E}_{1}$-structure $+_{J}$,
give paths in $\Theta(M\times J)_{0}$

$\displaystyle\tau_{WW}(\mathrm{alex}(\rho))$
$\displaystyle=\tau_{WW}(\mathrm{alex}^{s}(\rho+_{J}p_{-[\rho]}))+_{J}\tau_{WW}(\phi_{[\rho]})$
$\displaystyle\sim\mathrm{alex}^{s}(\iota_{H}(\rho)+_{J}\iota_{H}(p_{-[\rho]}))+_{J}\phi_{(-1)^{d}\overline{[\rho]}}$
$\displaystyle\sim\mathrm{alex}^{s}(\iota_{H}(\rho)+_{J}p_{-[\iota_{H}(\rho)]})+_{J}\phi_{[\iota_{H}(\rho)]}=:\mathrm{alex}(\iota_{H}(\rho))$

which depend continuously in $\rho\in H(M\times J)$. This defines the required
equivalence $\tau_{WW}\circ\mathrm{alex}\simeq\mathrm{alex}\circ\iota_{H}$.

The canonical homotopy between the map $(-)+_{J}*$ and the indentity (in both
spaces) gives the homotopy between $\mathrm{alex}^{s}$ and the composition
$H^{s}(M)\xhookrightarrow{}H(M)\overset{\mathrm{alex}}{\longrightarrow}\Theta(M\times
J)_{0}$. Therefore $\pi_{>0}(\mathrm{alex})=\pi_{>0}(\mathrm{alex}^{s})$ is an
isomorphism. Now by construction, under the identifications $\pi_{0}(H(M\times
J))\cong\mathrm{Wh}(\pi)$ and $\pi_{0}(\Theta(M\times
J)_{0})\cong\mathrm{Wh}(\pi)$, the map $\pi_{0}(\mathrm{alex})$ is the
identity of $\mathrm{Wh}(\pi)$. It follows then that $\mathrm{alex}$ is indeed
an equivalence. ∎

###  5.3. From $h$-cobordism spaces back to $A$-theory

Given a spherical fibration $\xi$ over $M$ equipped with a section, fibrewise
smashing a retractive space over $M$ with $\xi$ gives rise to a functor
$-\cdot\xi:\mathbf{A}(M)\to\mathbf{A}(M)$ which, by [Vog85, Prop. 2.5], makes
the following diagram homotopy commutative:

(5.11)
${\mathbf{A}(M)}$${\mathbf{A}(M)}$${\mathbf{A}(M),}$$\scriptstyle{\tau_{\epsilon}}$$\scriptstyle{\tau_{\xi}}$$\scriptstyle{-\cdot\xi}$

where $\epsilon:=\epsilon^{0}=M\times S^{0}$ is the trivial $0$-dimensional
sphere bundle over $M$. When $\xi=\epsilon^{d}=M\times S^{d}$ is the trivial
$d$-spherical fibration, the functor $-\cdot\xi$ corresponds to
$\Sigma^{d}_{M}(-):\mathbf{A}(M)\to\mathbf{A}(M)$, the $d$-fold fibrewise
suspension over $M$ (cf. [Vog85, p. 281]). By the additivity theorem of
[Wal85, Prop. 1.6.2] applied to the Waldhausen category of retractive spaces
over $M$, it follows that $\Sigma^{d}_{M}$ acts (up to homotopy) as $(-1)^{d}$
on $\mathbf{A}(M)$, and thus by (5.11)

(5.12) $\xi=M\times S^{d}\implies\mathbf{A}(M;\xi)\approx
S^{d\cdot(\sigma-1)}\wedge\mathbf{A}(M;\epsilon).$

Vogell also defined [Vog85, p. 299] a homotopy involution444Vogell refers to
$\mathscr{T}$ as a _weak_ involution in the sense that it is a homotopy
involution when restricted to ”any compactum” (cf. [Vog85, Lem. 2.4]) or, in
better words, to each stage of the colimit in [Vog85, p. 299] modelling
$A(M)$. $\mathscr{T}$ on $A(M)=\Omega^{\infty}\mathbf{A}(M)$, compatible with
the splitting $A(M)\simeq\mathrm{Wh}^{\mathrm{Diff}}(M)\times Q_{+}M$, such
that $\Omega\mathscr{T}$ extends $\iota_{H}$ on
$H(M)\subset\mathcal{H}(M)\simeq\Omega\mathrm{Wh}^{\mathrm{Diff}}(M)$. Here
$\mathcal{H}(M):=\operatorname{\operatorname{hocolim}}_{k}\ H(M\times I^{k})$
is the stable $h$-cobordism space of $M$, which is equivalent to
$\Omega\mathrm{Wh}(M)$ by the stable parametrised $h$-cobordism theorem of
Waldhausen–Jahren–Rognes [WJR13, Thm. 0.1]. Vogell further showed in [Vog85,
Cor. 2.10] that $\mathscr{T}$ and the involution $\tau_{\xi}$ agree up weak
homotopy when $\xi$ is the $d$-spherical fibration associated to the stable
tangent bundle $TM\oplus\epsilon^{1}$ of $M^{d}$. The upshot then is that when
$M^{d}$ is stably paralellisable, the Weiss–Williams involution is compatible
with $(-1)^{d}\tau_{\epsilon}$ in the following sense.

######  Theorem 5.11.

If $M$ is stably paralellisable, then there is an equivalence away from two

$\Omega^{\infty}\big{(}\Theta(M)_{hC_{2}}\big{)}\simeq_{[\frac{1}{2}]}\Omega^{\infty}\big{(}S^{d\cdot(\sigma-1)-1}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M;\epsilon)_{hC_{2}}\big{)}.$

######  Remark 5.12.

The equivalence above is not stated to be one of infinite loop spaces (even
though it most probably is). Moreover, the weaker statement that the
equivalence in Theorem 5.11 holds only after taking Postnikov sections
$\tau_{\scaleobj{0.85}{\geq}N}(-)$ for any $N\geq 0$ (and maybe even replacing
$\Theta(-)$ and $\mathbf{Wh}^{\mathrm{Diff}}(-)$ by $\Theta^{s}(-)$ and
$\mathbf{Wh}^{\mathrm{Diff}}_{\hskip 2.13394pts}(-)$) requires a much simpler
proof than the one we will give below; namely, one does not need Construction
5.9, Lemma 5.14 and Claim 2 anymore. Even though this weaker version will be
enough for the proof of Theorem B, we prefer to state and proof Theorem 5.11
in this generality as the reader may find it of independent interest. We
encourage the eager reader to figure out the details in the weaker case.

We will need the a few preliminary results for the proof of Theorem 5.11.

######  Lemma 5.13.

For each $k\geq 0$, there is a natural $C_{2}$-equivariant equivalence of
spectra

(5.13) $e_{k}:\Theta(M\times I^{k})\simeq
S^{k\cdot(\sigma-1)}\wedge\Theta(M).$

such that the following square is homotopy commutative:

${\Theta(M\times I^{k})}$${S^{\sigma-1}\wedge\Theta(M\times
I^{k-1})}$${S^{k\cdot(\sigma-1)}\wedge\Theta(M)}$${S^{k\cdot(\sigma-1)}\wedge\Theta(M).}$

$\scriptstyle\sim$

$\scriptstyle{e_{k}}$$\scriptstyle{\sim}$$\scriptstyle{e_{1}}$

$\scriptstyle\sim$

$\scriptstyle{{S}^{\sigma-1}\wedge e_{k-1}}$

###### Proof.

We may assume $k=1$, so that the above square is homotopy commutative by
construction. Recall
${\scaleobj{0.97}{\mathbb{R}}}^{a,b}:={\scaleobj{0.97}{\mathbb{R}}}^{a}\oplus
b\cdot\sigma$, and for any orthogonal functor $F(-)$ let $C_{2}=O(1)$ act on
$F({\scaleobj{0.97}{\mathbb{R}}}^{a,b})$ by the induced action. Finally let
$B(-):=B\mathrm{Diff}^{b}_{\partial}(M\times(-))$ and
$BI(-):=B\mathrm{Diff}^{b}_{\partial}(M\times I\times(-))$. The Alexander
trick-like map of [WW88, Prop. 1.5] is a $C_{2}$-equivariant map

$\mathrm{alex}:\mathrm{Diff}_{\partial}^{b}(M\times
I\times{\scaleobj{0.97}{\mathbb{R}}}^{a,b})\overset{\sim}{\longrightarrow}\Omega\mathrm{Diff}_{\partial}^{b}(M\times{\scaleobj{0.97}{\mathbb{R}}}^{a+1,b})$

which, upon delooping, gives rise to a $C_{2}$-equivariant equivalence on
basepoint components
$B(\mathrm{alex.}):BI({\scaleobj{0.97}{\mathbb{R}}}^{a,b})\simeq_{0}\Omega
B({\scaleobj{0.97}{\mathbb{R}}}^{a+1,b})$. Writing $\Xi$ for the
$C_{2}$-spectrum whose $n$-th space is
$B^{(1)}({\scaleobj{0.97}{\mathbb{R}}}^{1,n+1})$ and with stabilisation maps
$s_{0,1}:S^{1}\wedge B^{(1)}({\scaleobj{0.97}{\mathbb{R}}}^{1,n})\to
B^{(1)}({\scaleobj{0.97}{\mathbb{R}}}^{1,n+1})$, we obtain a
$C_{2}$-equivariant equivalence of spectra

(5.14) $B(\mathrm{alex.}):\Theta(M\times
I):=\Theta(BI)^{(1)}\overset{\sim}{\longrightarrow}\Xi:=\\{B^{(1)}({\scaleobj{0.97}{\mathbb{R}}}^{1,n+1})\\}_{n\geq
0}.$

But now the stabilisation map $s_{1,0}:S^{\sigma}\wedge
B^{(1)}({\scaleobj{0.97}{\mathbb{R}}}^{0,n})\to
B^{(1)}({\scaleobj{0.97}{\mathbb{R}}}^{1,n})=\Xi_{n-1}$ induces another
$C_{2}$-equivariant equivalence of spectra

$S^{\sigma-1}\wedge\Theta B^{(1)}\overset{\sim}{\longrightarrow}\Xi.$

Composing these two equivalences gives the one in the statement. ∎

Vogell introduced in [Vog85, p. 298] the lower and upper stabilisation maps
$\Sigma_{\ell},\Sigma_{u}:H(M)\to H(M\times I)$. Roughly, the former sends a
partition $\rho=(W,F,V)$ to $(U(W),W\cup_{F}W,\overline{M\times I\times
I\setminus U(W)})$, where $U(W)$ is obtained from $W$ by bending $W\times I$
into a $U$-shape, whilst $\Sigma_{u}$ does the same to $V$ instead of $W$ (see
Figure 7 for a pictorial representation of $\Sigma_{\ell}$). We will only be
interested in the lower stabilisation $\Sigma_{\ell}$, which we will denote by
$\Sigma$ for simplicity. Here’s how it interacts with the $h$-cobordism
involution $\iota_{H}$.

######  Lemma 5.14.

Let $+_{I}$ stand for the ”stacking in the $I$-direction”
$\mathbb{E}_{1}$-structure in $H(M\times I)$. Then:

$(a)\hskip 3.0pt\iota_{H}\Sigma+_{I}\Sigma\iota_{H}\simeq*:H(M)\to H(M\times
I),\qquad(b)\hskip 3.0pt\iota_{H}\Sigma^{2}\simeq\Sigma^{2}\iota_{H}:H(M)\to
H(M\times I^{2}).$

###### Proof.

We defer the proof of ($a$) to Lemma B.1 in Appendix B as it is a bit
technical. Then $(b)$ follows from

$\iota_{H}\Sigma^{2}\simeq\iota_{H}\Sigma^{2}+_{I}\Sigma(\iota_{H}\Sigma+_{I}\Sigma\iota_{H})\simeq(\iota_{H}\Sigma+_{I}\Sigma\iota_{H})\Sigma+_{I}\Sigma^{2}\iota_{H}\simeq\Sigma^{2}\iota_{H}.$

∎

We are now ready to deal with Theorem 5.11. If the reader has decided to skip
Construction 5.9, then they should replace in the argument below the symbols
$\mathrm{alex}$, $H(-)$, $\mathcal{H}(-)$, $\mathbf{Wh}^{\mathrm{Diff}}(-)$
and $\Theta(-)$ by $\mathrm{alex}^{s}$, $H^{s}(-)$, $\mathcal{H}^{s}(-)$,
$\mathbf{Wh}^{\mathrm{Diff}}_{\hskip 2.13394pts}(-)$ and
$\Theta^{s}(-):=\tau_{\scaleobj{0.85}{\geq}1}\Theta(-)$, respectively.

###### Proof of Theorem 5.11.

We may assume without loss of generality that $M$ is of the form
$M^{\prime}\times J$ for some $(d-1)$-manifold $M^{\prime}$ with $d\geq 5$:
indeed as there is a homotopy $C_{2}$-equivariant equivalence $S^{2}\approx
S^{2\sigma}$, it follows by Lemma 5.13 and Proposition 5.3 that there are
equivalences of spectra

$\displaystyle\Theta(M\times J^{2k})_{hC_{2}}$
$\displaystyle\simeq_{[\frac{1}{2}]}\Theta(M)_{hC_{2}},$
$\displaystyle\big{(}S^{(d+2k)\cdot(\sigma-1)-1}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M\times
J^{2k};\epsilon)\big{)}_{hC_{2}}$
$\displaystyle\simeq_{[\frac{1}{2}]}\big{(}S^{d\cdot(\sigma-1)-1}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M;\epsilon)\big{)}_{hC_{2}}.$

In the second equivalence we also use that $\mathbf{Wh}^{\mathrm{Diff}}(-)$
and $\tau_{\epsilon}$ are homotopy invariant. We will not explicitly use the
$J$-intervals nor that $d\geq 5$ anymore, which we only need for the map
”$\mathrm{alex}$” of Construction 5.9 to be well-defined. The intervals $I$
below should not be confused with the $J$ one (present only in disguise).

Let $k\geq 0$ and let $\xi$ denote the $(d+k)$-spherical fibration
corresponding to the stable tangent bundle $T(M\times
I^{k})\oplus\epsilon^{1}$. If $M$ (and hence $M\times I^{k}$) is stably
parallelisable, the involution $(-1)^{d+k}\tau_{\epsilon}$ is homotopic to
$\tau_{\xi}$ by (5.12), and by [Vog85, Prop. Cor. 2.10] it agrees with
$\mathscr{T}$ (and hence extends $\iota_{H}$). All in all, we obtain a zig-zag
of homotopy $C_{2}$-equivariant maps

(5.15) ${\Omega^{\infty}\Theta(M\times I^{k})}$${H(M\times
I^{k})}$${\Omega^{\infty}\big{(}S^{(d+k)\cdot(\sigma-1)-1}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M\times
I^{k})\big{)}}$${\Omega^{\infty}(S^{k\cdot(\sigma-1)}\wedge\Theta(M))}$${\Omega^{\infty}\big{(}S^{(d+k)\cdot(\sigma-1)-1}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M)\big{)}}$

$\simeq$

(5.13)

$\scriptstyle{(\ref{alexHcobmap})}$$\scriptstyle{\mathrm{alex}}$$\scriptstyle{(\dagger)}$

$\simeq$

Every space involved in (5.15) is an $\mathbb{E}_{1}$-group if $k\geq 1$: both
$H^{s}(M\times I^{k})$ and $\Omega^{\infty}(\Theta(M\times I^{k}))$ by
stacking in the first coordinate of $I^{k}$, and the others by their own
infinite loop structures.

######  Claim 1.

All of the maps in (5.15) are $H$-maps if $k\geq 1$.

###### Proof.

The only maps that are not obviously $H$-maps are (5.13) and ($\dagger$). For
the former, we may assume that $k=1$, and we need only show that
$\Omega^{\infty}(\Theta(M\times I))\to\Omega^{\infty}\Xi$ of (5.14) is an
$H$-map, where $\Xi$ is as in the proof of Lemma 5.13. Non-equivariantly, such
map is the colimit as $n$ goes to infinity of

$\mathrm{alex}:\Omega^{n}\left(\frac{\mathrm{Diff}^{b}_{\partial}(M\times
I\times{\scaleobj{0.97}{\mathbb{R}}}^{n+1})}{\mathrm{Diff}^{b}_{\partial}(M\times
I\times{\scaleobj{0.97}{\mathbb{R}}}^{n})}\right)\overset{\sim}{\longrightarrow}\Omega^{n+1}\left(\frac{\mathrm{Diff}^{b}_{\partial}(M\times{\scaleobj{0.97}{\mathbb{R}}}^{n+2})}{\mathrm{Diff}^{b}_{\partial}(M\times
I\times{\scaleobj{0.97}{\mathbb{R}}}^{n+1})}\right).$

One verifies that stacking in the $I$-direction of the domain of this map
corresponds, up to homotopy in the codomain, to composing loops in the
$(n+1)$-st coordinate. So (5.13) is indeed an $H$-map.

Non equivariantly, the map $(\dagger)$ is the composition $(\dagger):H(M\times
I^{k})\xhookrightarrow{}\mathcal{H}(M\times
I^{k})\simeq\Omega^{\infty+1}\mathbf{Wh}^{\mathrm{Diff}}(M\times I^{k})$,
where the last equivalence is the stable parametrised $h$-cobordism theorem of
Waldhausen–Jahren–Rognes [WJR13, Thm. 0.1]. As communicated to us in private
by Bjørn Jahren and John Rognes, such equivalence is only stated to hold in
the category of topological spaces (and not of infinite loop spaces, though it
should definitely also hold there). We now explain why this equivalence is one
of $H$-groups (which is the general consensus, but we couldn’t find it written
down anywhere): again assume $k=1$. One reduces to the $PL$-case as in [WJR13,
pp. 15 & 16]. Then if $X$ is a simplicial set such that $|X|\simeq M$, the
equivalence in the $PL$-setting is induced by a zig-zag of equivalences of
simplicial sets (cf. the left vertical column of [WJR13, Eq. (0.4)])

${\mathcal{H}(M\times I)}$${\cdot}$${s\mathscr{C}^{h}(X\times
I),}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where $s\mathscr{C}^{h}(X\times I)$ is the category (seen as a simplicial set
by taking its nerve) of acyclic cofibrations $X\times I\xhookrightarrow{}Y$
together with simple maps over $X\times I$. One can verify that it makes sense
to stack in the $I$-direction in each of the simplicial sets involved in the
zig-zag, and that the maps between them respect this monoidal structure. Let
$\mu_{0}:s\mathscr{C}^{h}(X\times I)\times s\mathscr{C}^{h}(X\times I)\to
s\mathscr{C}^{h}(X\times I)$ stand for this monoidal structure, given
explicitly by $\mu_{0}(Y,Z):=Y{}_{X\times{1}}\cup_{X\times 0}Z$. There is
another monoidal structure $\mu_{1}$ induced by the pushout along $X\times I$,
i.e. $\mu_{1}(Y,Z):=Y\cup_{X\times I}Z$. Sliding gives a homotopy between
$\mu_{0}$ and $\mu_{1}$: intuitively, the maps

$\mu_{t}:s\mathscr{C}^{h}(X\times I)\times s\mathscr{C}^{h}(X\times
I)\longrightarrow s\mathscr{C}^{h}(X\times I),\quad(Y,Z)\longmapsto
Y{}_{X\times[1-t,1]}\cup_{X\times[0,t]}Z,\quad t\in[0,1],$

constitute the homotopy. More precisely, consider the simplicial category
$s\widetilde{\mathscr{C}}^{h}(X)$ [WJR13, Defn. 3.1.1] whose objects in
simplicial degree $q$ consist of commutative diagrams

${X\times\Delta^{q}}$${Y}$${\Delta^{q},}$$\scriptstyle{\mathrm{pr}}$$\scriptstyle{i}$$\scriptstyle{\sim}$$\scriptstyle{\pi}$

where $i$ is an acyclic cofibration and $\pi$ is a Serre fibration. The
inclusion
$s\mathscr{C}^{h}(X)\xhookrightarrow{}s\widetilde{\mathscr{C}}^{h}(X)$ as the
$0$-simplices is a homotopy equivalence by [WJR13, Cor. 3.5.2], where a
simplicial category is seen as a bisimplicial set by taking its nerve, and a
bisimplicial set as an ordinary simplicial set by taking its totalisation. The
monoidal structures $\mu_{t}$ make perfectly good sense in
$s\widetilde{\mathscr{C}}^{h}(X\times I)$, and one can indeed define a
simplicial homotopy between $\mu_{0}$ and $\mu_{1}$ in this setting resembling
the idea above.

But by Proposition 3.1.1, and Theorems 3.1.7 and 3.3.1 of [Wal85] (see also
[WJR13, p. 5]), for any simplicial set $T$, there is a zig-zag of equivalences
connecting $|s\mathscr{C}^{h}(T)|$ and
$\Omega\mathrm{Wh}^{PL}(T):=\Omega^{\infty+1}(\mathbf{Wh}^{PL}(T))$ which is
monoidal up to homotopy with respect to $\mu_{1}$ in the domain and the loop
structure on the looped Whitehead space. It hence follows that ($\dagger$) is
indeed a zig-zag of $H$-maps. ∎

######  Claim 2.

For each $k\geq 1$, the diagram

(5.16) ${\Omega^{\infty}(S^{2k\cdot(\sigma-1)}\wedge\Theta(M))}$${H(M\times
I^{2k})}$${\Omega^{\infty}(S^{(d+2k)\cdot(\sigma-1)-1}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M))}$${\Omega^{\infty}(S^{(2k+2)\cdot(\sigma-1)}\wedge\Theta(M))}$${H(M\times
I^{2k+2})}$${\Omega^{\infty}(S^{(d+2k+2)\cdot(\sigma-1)-1}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M))}$

$\approx$

$\scriptstyle{\Sigma^{2}}$

$\approx$

is homotopy commutative, where the rows are the zig-zags (5.15) and the
vertical external maps are induced by the homotopy $C_{2}$-equivariant
equivalence $\mathbb{S}^{0}\approx\mathbb{S}^{2\cdot(\sigma-1)}$.

###### Proof.

The right hand square is clearly commutative. The commutativity of the left
one will be a consequence of the commutativity of the one in Lemma 5.13 and
the commutativity of the (non-equivariant) square

${H(M\times I^{k})}$${\Theta(M\times
I^{k})_{0}}$${\Omega^{k}\Theta(M)_{k}}$${H(M\times I^{k+1})}$${\Theta(M\times
I^{k+1})_{0}}$${\Omega^{k+1}\Theta(M)_{k+1}.}$$\scriptstyle{\Sigma}$$\scriptstyle{\mathrm{alex}}$[WW88,
Prop. 1.5]$\scriptstyle{s^{\vee}}$$\scriptstyle{\mathrm{alex}}$[WW88, Prop.
1.5]

This last square is commutative by a delooped version of the proof of [WW88,
Prop. 1.12] (compare with Proposition 3.8). ∎

Clearly $\Sigma^{2}$ is an $H$-map and also homotopy $C_{2}$-equivariant (with
respect to $\iota_{H}$) by Lemma 5.14($b$). Therefore by Claim 1, all of the
maps involved in (5.16) are $H$-maps and homotopy $C_{2}$-equivariant. Taking
the homotopy colimit as $k\to\infty$, we obtain a homotopy $C_{2}$-equivariant
zig-zag

(5.17)
${\Omega^{\infty}(\Theta(M))}$${\underset{k}{\operatorname{\operatorname{hocolim}}}\
H(M\times
I^{2k})}$${\Omega^{\infty}(S^{d\cdot(\sigma-1)-1}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M))}$$\scriptstyle{\approx}$$\scriptstyle{\approx}$

of $H$-maps. The connectivity of, say, the upper horizontal maps in (5.16) is
$\phi(d+2k)\gtrsim(d+2k)/3$ by Igusa’s theorem and, as this lower bound
increases linearly with $k$, the horizontal maps in (5.17) are indeed
equivalences. The equivalence in the statement now follows by the second part
of Proposition 5.3 applied to (5.17), and because taking homotopy
$C_{2}$-orbits commutes up to equivalence with $\Omega^{\infty}(-)$ if $2$ is
inverted (as in Corollary 5.4). The proof of Theorem 5.11 is now complete. ∎

######  Corollary 5.15.

If $M^{d}$ is stably paralellisable and $P\subset M^{d}$ is a codimension zero
submanifold with $p\leq d-3$ (in the notation of Theorem A), then there is an
equivalence away from two

$\Omega^{\infty}\big{(}\Sigma^{-1}\Theta(M,\overline{M-P})_{hC_{2}}\big{)}\simeq_{[\frac{1}{2}]}\Omega^{\infty}\big{(}\big{(}S^{d\cdot(\sigma-1)-2}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M,M-P;\epsilon)\big{)}_{hC_{2}}\big{)}.$

###### Proof.

Note that $\overline{M-P}$ is stably parallelisable because $M$ is. The zig-
zag (5.17) is functorial with respect to codimension zero embeddings of stably
paralellisable manifolds, and hence taking the homotopy fibre of the map
between $(\ref{zigzaginvolutions})$ applied to $\overline{M-P}$ and
$(\ref{zigzaginvolutions})$ applied to $M$, we obtain another homotopy
$C_{2}$-equivariant zig-zag of equivalences

${\Omega^{\infty}(\Sigma^{-1}\Theta(M,\overline{M-P}))}$${\underset{k}{\operatorname{\operatorname{hocolim}}}\
C\mathrm{Emb}(P\times I^{2k},M\times
I^{2k})}$${\Omega^{\infty}(S^{d\cdot(\sigma-1)-2}\wedge\mathbf{Wh}^{\mathrm{Diff}}(M,M-P)).}$$\scriptstyle{\approx}$$\scriptstyle{\approx}$

Note that $M-P\to M$ is $2$-connected because $p\leq d-3$, and hence

${\Theta(\overline{M-P})}$${\Theta^{s}({M})}$${\Theta(M)}$${\Theta^{s}(M)}$

is cartesian (and similarly for $H(-)$ and $\mathbf{Wh}^{\mathrm{Diff}}(-)$ in
place of $\Theta(-)$). So, indeed as mentioned before, one gets the same zig-
zag if dealing only with the simple version of Theorem 5.11 (so as to skip
Construction 5.9). In any case, the claim now follows as before by applying
Proposition 5.3 to the zig-zag above. ∎

###  5.4. The canonical involution in algebraic $K$-theory

We now define the canonical involution $\tau_{\epsilon}$ on $A(X)$ and explore
its connection to an involution in the model of $A$-theory via ”spaces of
matrices with values in the ring up to homotopy” $Q_{+}\Omega X$ [Wal85,
§2.2]. All throughout, let $G:=GX$ denote the topological monoid of Moore
loops on $X$, and write $\mathbb{S}[G]$ for the $\mathbb{E}_{1}$-ring spectrum
$\mathbb{S}\wedge G_{+}$.

We will work over the $\infty$-category $\mathsf{Mod}_{\mathbb{S}[G]}$ of
right $\mathbb{S}[G]$-module spectra; we will also write
${}_{\mathbb{S}[G]}\mathsf{Mod}$ for the $\infty$-category of left
$\mathbb{S}[G]$-modules. Then for $m\geq 1$, if
$\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])$ denotes the homotopy invertible
components of the mapping space
$\mathsf{Mod}_{\mathbb{S}[G]}(\oplus^{m}\mathbb{S}[G],\oplus^{m}\mathbb{S}[G])$,
Waldhausen showed in [Wal85, Thm. 2.2.1] that for $X$ connected, there is a
natural equivalence

(5.18)
$A(X)\simeq{\scaleobj{0.97}{\mathbb{Z}}}\times\underset{m}{\operatorname{\operatorname{hocolim}}}\
B\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])^{+}.$

In order to define $\tau_{\epsilon}$, we will introduce compatible anti-
involutions on $\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])$ defined in terms of
Spanier–Whitehead duality. As $\mathbb{S}[G]$ is not commutative, this duality
really arises as an instance of a duality in the symmetric closed bicategory
$\mathsf{Bimod}{S}$ of bimodule spectra, in the sense of [MS06, §16.4]. This
duality coincides with the one considered by Vogell in [Vog85, §1].

First observe that a right $\mathbb{S}[G]$-module $M$ can always be regarded
as a left $\mathbb{S}[G]$-module by

${\mathbb{S}[G]\otimes
M}$${M\otimes\mathbb{S}[G]}$${M\otimes\mathbb{S}[G^{\mathrm{op}}]=M\otimes\mathbb{S}[G]}$${M,}$swap$\scriptstyle{\mathrm{Id}_{M}\otimes\text{inv}}$$\scriptstyle{\mathrm{act}}$

where ”inv” stands for inversion in the monoid $G$–write $M_{\ell}$ for this
left $\mathbb{S}[G]$-module. Here $\otimes=\otimes{S}$ stands for the usual
smash product of spectra. Note also that
${}_{\mathbb{S}[G]}\mathsf{Mod}(M_{\ell},M_{\ell})\simeq\mathsf{Mod}_{\mathbb{S}[G]}(M,M)$
as $\mathbb{E}_{1}$-algebras. If $\nu:\mathbb{S}\to\mathbb{S}[G]_{\ell}$
denotes the unit, consider the map of spectra

${\eta_{1}:\mathbb{S}\simeq\mathbb{S}\otimes_{\mathbb{S}[G]}\mathbb{S}[G]}$${\mathbb{S}[G]\otimes_{\mathbb{S}[G]}\mathbb{S}[G]_{\ell}}$$\scriptstyle{\nu\otimes
1}$

and the map of $(\mathbb{S}[G],\mathbb{S}[G])$-bimodules

${I_{1}:\mathbb{S}[G]_{\ell}\otimes\mathbb{S}[G]}$${\mathbb{S}[G]\otimes\mathbb{S}[G]}$${\mathbb{S}[G].}$$\scriptstyle{\text{inv}\otimes
1}$act

Then $(\eta_{1},I_{1})$ make $(\mathbb{S}[G],\mathbb{S}[G]_{\ell})$ a dual
pair [MS06, Defn. 16.4.1]. More generally, the map of spectra

${\eta_{m}:\mathbb{S}}$${\bigoplus_{i,j=1}^{m}\mathbb{S}[G]\otimes_{\mathbb{S}[G]}\mathbb{S}[G]_{\ell}\cong\bigoplus^{m}_{j=1}\mathbb{S}[G]\otimes_{\mathbb{S}[G]}\big{(}\bigoplus^{m}_{i=1}\mathbb{S}[G]\big{)}_{\ell}}$$\scriptstyle{\bigoplus_{i,j}\delta_{ij}\eta_{1}}$

together with the map of $(\mathbb{S}[G],\mathbb{S}[G])$-bimodules

${I_{m}:\big{(}\bigoplus^{m}_{i=1}\mathbb{S}[G]\big{)}_{\ell}\otimes\bigoplus^{m}_{j=1}\mathbb{S}[G]\cong\bigoplus_{i,j=1}^{m}\mathbb{S}[G]_{\ell}\otimes\mathbb{S}[G]}$${\mathbb{S}[G],}$$\scriptstyle{\bigoplus_{i,j}\delta_{ij}I_{1}}$

exhibit $(\oplus^{m}\mathbb{S}[G])_{\ell}$ as a right dual to
$\oplus^{m}\mathbb{S}[G]$. Therefore as in [MS06, Prop. 16.4.9], $I_{m}$
induces an equivalence of left $\mathbb{S}[G]$-modules
$\widetilde{I}_{m}:(\oplus^{m}\mathbb{S}[G])_{\ell}\simeq
D_{r}(\oplus^{m}\mathbb{S}[G]):=\underline{\mathrm{Hom}}_{\mathbb{S}[G]}(\oplus^{m}\mathbb{S}[G],\mathbb{S}[G])$,
where the right hand side stands for the right $\mathbb{S}[G]$-linear mapping
spectrum.

With (5.18) and Lemma 5.5 in mind, the involution $\tau_{\epsilon}$ on
$A(X;\epsilon)$ is then induced by the map of $\mathbb{E}_{1}$-algebras

(5.19)
${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${{}_{G}\mathrm{Aut}(D_{r}(\oplus^{m}\mathbb{S}[G]))^{\mathrm{op}}\overset{\widetilde{I}_{\\#}}{\simeq}{}_{G}\mathrm{Aut}((\oplus^{m}\mathbb{S}[G])_{\ell})^{\mathrm{op}}\simeq\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])^{\mathrm{op}},}$$\scriptstyle{D_{r}}$

where $\widetilde{I}_{\\#}$ stands for conjugation with the equivalence
$\widetilde{I}_{m}$. It will be convenient to think of (5.19) in the following
way: let $GL_{m}(Q_{+}G)$ denote the union of path components in
$(Q_{+}G)^{m\times m}$ in the image of
$\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])$ under the natural equivalence

${u:\mathsf{Mod}_{\mathbb{S}[G]}(\oplus^{m}\mathbb{S}[G],\oplus^{m}\mathbb{S}[G])}$${\mathsf{Mod}_{\mathbb{S}[G]}(\oplus^{m}\mathbb{S}[G],\prod^{m}\mathbb{S}[G])\simeq\mathsf{Sp}(\mathbb{S},\mathbb{S}[G])^{m\times
m}\simeq(Q_{+}G)^{m\times m},}$$\scriptstyle{\sim}$

where $\mathsf{Sp}\simeq\mathsf{Mod}{S}$ stands for the $\infty$-category of
spectra. So $u:\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])\simeq GL_{m}(Q_{+}G)$
and, just as in standard linear algebra, under this equivalence the anti-
involution (5.19) corresponds to the rule that sends a matrix $A$ to its
conjugate transpose $A^{\dagger}$ (conjugate with respect to inversion of $G$
in $\mathbb{S}[G]$). More precisely:

######  Proposition 5.16.

Write
$\mathrm{End}_{G}(\oplus^{m}\mathbb{S}[G]):=\mathsf{Mod}_{\mathbb{S}[G]}(\oplus^{m}\mathbb{S}[G],\oplus^{m}\mathbb{S}[G])$
with the action of the cyclic group $C_{m}$ by conjugation with the
permutation automorphisms of $\oplus^{m}\mathbb{S}[G]$. Let $C_{m}$ act
similarly on $(Q_{+}G)^{m\times m}$ by conjugation. Then the following square
is commutative in the homotopy category of $C_{m}$-spaces:

(5.20)
${\mathrm{End}_{G}(\oplus^{m}\mathbb{S}[G])}$${\mathrm{End}_{G}(\oplus^{m}\mathbb{S}[G])}$${(Q_{+}G)^{m\times
m}}$${(Q_{+}G)^{m\times m}.}$$\scriptstyle{u}$

$\scriptstyle\sim$

$\scriptstyle{(\ref{tauepsilonantiinv})}$$\scriptstyle{\sim}$$\scriptstyle{u}$

$\scriptstyle\sim$

$\scriptstyle{\dagger}$$\scriptstyle{\sim}$

######  Remark 5.17.

Passing to the homotopy invertible components in (5.20), we obtain the
following commutative square in the homotopy category of $C_{m}$-spaces

${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${GL_{m}(Q_{+}G)}$${GL_{m}(Q_{+}G).}$$\scriptstyle{u}$

$\scriptstyle\sim$

$\scriptstyle{(\ref{tauepsilonantiinv})}$$\scriptstyle{\sim}$$\scriptstyle{u}$

$\scriptstyle\sim$

$\scriptstyle{\dagger}$$\scriptstyle{\sim}$

We have suppressed the $(-)^{\mathrm{op}}$ in the codomain of the map (5.19)
in the previous squares to emphasise that such squares take place in the
homotopy category of ($C_{m}$-)spaces, as opposed to that of
$\mathbb{E}_{1}$-spaces.

###### Proof of Proposition 5.16.

First note that all the maps involved in (5.20) are indeed $C_{m}$-maps: the
only one that is not obviously so is (5.19), but this follows from the
observation that $I_{m}$ is $C_{m}$-equivariant for the diagonal action on the
domain and the trivial action on the target. Note also that the $C_{m}$-action
on $(Q_{+}G)^{m\times m}$ restricts to a cofree $C_{m}$-action on each of the
right $C_{m}$-cosets of the diagonal subspace, and hence $(Q_{+}G)^{m\times
m}=\prod^{m}\mathrm{coInd}_{e}^{C_{m}}Q_{+}G$ as a $C_{m}$-space. Thus, in
order to show that (5.20) commutes in the homotopy category of $C_{m}$-spaces,
it suffices to prove that it commutes in the homotopy category of spaces after
postcomposing it with the map

$(Q_{+}G)^{m\times
m}=\prod^{m}\mathrm{coInd}_{e}^{C_{m}}Q_{+}G\longrightarrow\prod^{m}Q_{+}G$

that records the first column of a matrix.

Now given an endomorphism $h$ of $\oplus^{m}\mathbb{S}[G]$, the $(m\times
m)$-matrix $u(h)=(h_{ij})\in GL_{m}(Q_{+}G)$ has components

${h_{ij}:\mathbb{S}}$${\mathbb{S}[G]}$${\bigoplus_{k=1}^{m}\mathbb{S}[G]}$${\bigoplus_{k=1}^{m}\mathbb{S}[G]}$${\mathbb{S}[G].}$$\scriptstyle{\nu}$$\scriptstyle{\mathrm{inc}_{j}}$$\scriptstyle{h}$$\scriptstyle{\mathrm{pr}_{i}}$

Slightly abusing the notation, we will write $\tau_{\epsilon}$ to mean (5.19).
Then we must only check that $\tau_{\epsilon}(h)_{ij}$ is homotopic to
$\overline{h}_{ji}$, coherently in $h$ (and for $j=1$, though it is still true
for all $j$ of course). Observe now that
$(-)_{\ell}:\mathsf{Mod}_{\mathbb{S}[G]}\to{}_{\mathbb{S}[G]}\mathsf{Mod}$ is
a functor over $\mathsf{Sp}$, and hence the last equivalence in (5.19) happens
over the automorphism space of $\oplus^{m}\mathbb{S}[G]$ as a regular
spectrum. Consequently, $\tau_{\epsilon}(h)_{ij}$ is by definition the top
horizontal composition in the diagram of spectra

${\mathbb{S}}$${\mathbb{S}[G]_{\ell}}$${\bigoplus_{k=1}^{m}\mathbb{S}[G]_{\ell}}$${\bigoplus_{k=1}^{m}\mathbb{S}[G]_{\ell}}$${\mathbb{S}[G]_{\ell}}$${\mathbb{S}[G]}$${\mathbb{S}[G]}$${\underline{\mathrm{Hom}}_{\mathbb{S}[G]}(\mathbb{S}[G],\mathbb{S}[G])}$${\underline{\mathrm{Hom}}_{\mathbb{S}[G]}(\oplus^{m}\mathbb{S}[G],\mathbb{S}[G])}$${\underline{\mathrm{Hom}}_{\mathbb{S}[G]}(\oplus^{m}\mathbb{S}[G],\mathbb{S}[G])}$${\underline{\mathrm{Hom}}_{\mathbb{S}[G]}(\mathbb{S}[G],\mathbb{S}[G]).}$$\scriptstyle{\nu}$$\scriptstyle{\nu}$${(*_{1})}$$\scriptstyle{\mathrm{inv}}$

$\scriptstyle\sim$

$\scriptstyle{\mathrm{inc}_{j}}$$\scriptstyle{\widetilde{I}_{m}}$

$\scriptstyle\sim$

${(*_{2})}$$\scriptstyle{\tau_{\epsilon}(h)}$${(*_{3})}$$\scriptstyle{\widetilde{I}_{m}}$

$\scriptstyle\sim$

$\scriptstyle{\mathrm{pr}_{i}}$

$\simeq$

$\simeq$

$\scriptstyle{\mathrm{inv}}$

$\scriptstyle\sim$

$\scriptstyle{\mathrm{pr}_{j}^{*}}$$\scriptstyle{h^{*}}$$\scriptstyle{\mathrm{inc}_{i}^{*}}$

On the other hand, one recognises the composition that goes through the bottom
row to be $\overline{h}_{ji}$. Note that the left triangle is commutative, and
that the square $(*_{2})$ is too by definition of $\tau_{\epsilon}(h)$.
Moreover, ($*_{1}$) and $(*_{3})$ do not depend on $h$; we must then argue
that ($*_{1}$) and $(*_{3})$ are commutative up to homotopy.

For the commutativity of $(*_{1})$, first observe that the left vertical
composition of $(*_{1})$ coincides up to homotopy with the equivalence of left
$\mathbb{S}[G]$-modules
$\widetilde{I}_{1}:\mathbb{S}[G]_{\ell}\simeq\underline{\mathrm{Hom}}_{\mathbb{S}[G]}(\mathbb{S}[G],\mathbb{S}[G])$.
This is because, under the usual tensor-hom adjunction, both maps represent
the same element in

$\pi_{0}\left({}_{\mathbb{S}[G]}\mathsf{Mod}\big{(}\mathbb{S}[G]_{\ell},\underline{\mathrm{Hom}}_{\mathbb{S}[G]}(\mathbb{S}[G],\mathbb{S}[G])\big{)}\right)\cong\pi_{0}\big{(}{}_{\mathbb{S}[G]}\mathsf{Mod}_{\mathbb{S}[G]}\big{(}\mathbb{S}[G]_{\ell}\otimes\mathbb{S}[G],\mathbb{S}[G]\big{)}\big{)}$

by definition of $I_{1}$. But now ($*_{1}$), with $\widetilde{I}_{1}$ in place
of the left vertical composition, commutes up to homotopy as both compositions
represent the same element in

$\pi_{0}\left({}_{\mathbb{S}[G]}\mathsf{Mod}\big{(}\mathbb{S}[G]_{\ell},\underline{\mathrm{Hom}}_{\mathbb{S}[G]}(\oplus^{m}\mathbb{S}[G],\mathbb{S}[G])\big{)}\right)\cong\pi_{0}\big{(}{}_{\mathbb{S}[G]}\mathsf{Mod}_{\mathbb{S}[G]}\big{(}\mathbb{S}[G]_{\ell}\otimes(\oplus^{m}\mathbb{S}[G]),\mathbb{S}[G]\big{)}\big{)}$

simply because the following diagram commutes by definition of $I_{1}$ and
$I_{m}$:

${\mathbb{S}[G]_{\ell}\otimes\bigoplus^{m}\mathbb{S}[G]}$${\bigoplus^{m}\mathbb{S}[G]_{\ell}\otimes\bigoplus^{m}\mathbb{S}[G]}$${\mathbb{S}[G]_{\ell}\otimes\mathbb{S}[G]}$${\mathbb{S}[G].}$$\scriptstyle{1\otimes\mathrm{pr}_{j}}$$\scriptstyle{\mathrm{inc}_{j}\otimes
1}$$\scriptstyle{I_{m}}$$\scriptstyle{I_{1}}$

Finally the commutativity of ($*_{3}$) follows by a similar reasoning using
that

${\bigoplus^{m}\mathbb{S}[G]_{\ell}\otimes\mathbb{S}[G]}$${\mathbb{S}[G]_{\ell}\otimes\mathbb{S}[G]}$${\bigoplus^{m}\mathbb{S}[G]_{\ell}\otimes\bigoplus^{m}\mathbb{S}[G]}$${\mathbb{S}[G]}$$\scriptstyle{1\otimes\mathrm{inc}_{i}}$$\scriptstyle{\mathrm{pr}_{i}\otimes
1}$$\scriptstyle{I_{1}}$$\scriptstyle{I_{m}}$

is also commutative by definition. ∎

######  Warning 5.18.

The canonical involution $\tau_{\epsilon}$ induces an involution on the
Whitehead group

$\mathrm{Wh}(X):=\pi_{1}^{s}(\mathbf{Wh}^{\mathrm{Diff}}(X))\cong
GL({\scaleobj{0.97}{\mathbb{Z}}}[\pi_{1}(X)])^{\mathrm{ab}}/(\pm\pi_{1}(X)).$

In the foundational paper [Mil71], Milnor also defined an involution
$\mathrm{Wh}(X)\ni\kappa\mapsto\overline{\kappa}$ induced by sending a matrix
in $GL({\scaleobj{0.97}{\mathbb{Z}}}[\pi_{1}(X)])^{\mathrm{ab}}$ to its
conjugate transpose (conjugate with respect to inversion in $\pi_{1}(X)$).
This is an actual homomorphism because of the abelianisation present in the
general linear group of ${\scaleobj{0.97}{\mathbb{Z}}}[\pi_{1}(X)]$. It is
worth being aware that these two involutions on $\mathrm{Wh}(X)$ are only the
same after introducing a minus sign, i.e.

(5.21) $\tau_{\epsilon}(\kappa)=-\overline{\kappa}.$

This does not contradict the commutativity of (5.20). On the contrary, this
extra minus sign is the result of having to deloop the anti-involution (5.19)
in the sense of Lemma 5.5 in order to obtain $\tau_{\epsilon}$.

###  5.5. $A$-theory of a suspension

In this section we focus our attention on the homotopy type of $A(X;\epsilon)$
when $X$ is the suspension $\Sigma Y$ of a connected based space $Y$. By a
theorem of Carlsson–Cohen–Goodwillie–Hsiang555As pointed out in [BCC+96, p.
543], the proof in [CCGH87] has a serious flaw around p. 71. This issue was
fixed in [BCC+96, Cor. 4.15], and in particular the map $\theta$ of (5.22)
constructed in [CCGH87, §1] is still an equivalence. We are indebted to Tom
Goodwillie for his help in clearing out this matter and for carefully
explaining to us another more general principle for which (5.22) holds—namely,
it is the observation that if $F$ is a functor (from based spaces to based
spaces, say) whose $m$-th derivative spectrum is of the form $X\mapsto
X^{\wedge m}_{hC_{m}}$ for every $m\geq 1$, then its Taylor tower must split
globally. This is indeed the case for the functor $F(-):=\Omega\circ
A\circ\Sigma(-)$. [CCGH87, Thm. 3], in such cases there is an equivalence of
infinite loop spaces

(5.22) $\theta:\prod_{m\geq 1}Q(Y^{\wedge
m}_{hC_{m}})\overset{\sim}{\longrightarrow}\Omega\widetilde{A}(\Sigma Y),$

where $\widetilde{A}(-):=\operatorname{hofib}(A(-)\to A(*))$ and $C_{m}$ acts
on $Y^{\wedge m}$ by cyclic permutation of the factors. In this section, we
argue that (5.22) can be upgraded to be $C_{2}$-equivariant up to homotopy.

######  Proposition 5.19.

Let $Y$ be a connected, based $C_{2}$-space. There is an equivalence of
spectra

$\boldsymbol{\theta}:\bigvee_{m\geq
1}\Sigma^{\infty+\sigma}\big{(}(ED_{m})_{+}\wedge_{C_{m}}Y^{\wedge
m}\big{)}\overset{\sim}{\longrightarrow}\widetilde{\mathbf{A}}(\Sigma^{\sigma}Y;\epsilon)$

that is $C_{2}$-equivariant up to homotopy, and whose underlying (non-
equivariant) equivalence induces (5.22). Here
$\Sigma^{\sigma}Y:=S^{\sigma}\wedge Y$ and $D_{m}\subset\Sigma_{m}$ acts on
$Y^{\wedge m}$ by

$g\cdot(y_{1}\wedge\dots\wedge y_{m})\longmapsto g\cdot
y_{g(1)}\wedge\dots\wedge g\cdot y_{g(m)},\quad g\in D_{m},\quad y_{i}\in Y,$

where $Y$ is now seen as a based $D_{m}$-space (on which $C_{m}$ acts
trivially). Finally $C_{2}=D_{m}/C_{m}$ acts on
$(ED_{m})_{+}\wedge_{C_{m}}Y^{\wedge m}$ by its residual diagonal action.

######  Remark 5.20.

The $C_{2}$-space $\Sigma^{\sigma}Y$ induces an involution on
$\mathbf{A}(\Sigma^{\sigma}Y)$ which commutes with the canonical involution
$\tau_{\epsilon}$ described in the previous section, by naturality of its
construction. Therefore its composite gives the involution on
$\widetilde{\mathbf{A}}(\Sigma^{\sigma}Y;\epsilon)$ appearing in the statement
of Proposition 5.19. Alternatively, we can allow $X$ in the previous section
to mean a $C_{2}$-space (e.g. $\Sigma^{\sigma}Y$), and agree that
$\mathrm{inv}:G=GX\to G^{\mathrm{op}}$ there stands for inversion in the
monoid $G$ followed by the $C_{2}$-action on $X$.

In practice, we will apply Proposition 5.19 to the case when
$Y=S^{\sigma}\wedge Z$ for some trivial $C_{2}$-space $Z$, as then
$\Sigma^{\sigma}Y\simeq S^{2\sigma}\wedge Z\approx S^{2}\wedge Z$ because of
the homotopy $C_{2}$-equivariant equivalence $S^{2\sigma}\approx S^{2}$. In
such case, as $\mathbf{A}(-;\epsilon)$ is a homotopy functor, Proposition 5.19
provides a simple description of the homotopy $C_{2}$-equivariant homotopy
type of
$\widetilde{\mathbf{A}}(\Sigma^{2}Z;\epsilon)\approx\widetilde{\mathbf{A}}(\Sigma^{2\sigma}Z;\epsilon)$.
This, together with Proposition 5.3, can then be used to analyse the homotopy
type of $\widetilde{\mathbf{A}}(\Sigma^{2}Z;\epsilon)_{hC_{2}}$ away from $2$.

We will need the following observation for the proof of Proposition 5.19.

######  Lemma 5.21.

Let $X$ be a based, connected $C_{m}$-space. The equivalence

$\iota:B((C_{m}\ltimes\Omega X)^{\mathrm{op}})\simeq B(C_{m}\ltimes\Omega X)$

of Lemma 5.5 coincides up to equivalence with the delooping of the inversion
map

$\mathrm{inv}:(C_{m}\ltimes\Omega X)^{\mathrm{op}}\longrightarrow
C_{m}\ltimes\Omega
X,\quad(s^{i},\gamma)\mapsto(s^{-i},s^{i}\cdot\overline{\gamma}),$

where $\overline{\gamma}$ stands for the loop $\gamma$ with the reversed
orientation.

###### Proof.

Given a topological monoid $M$ equipped with a $C_{m}$-action, it is well-
known (see e.g. [AM04, §II, Thm. 1.12]) that the classifying space of the
semi-direct product $C_{m}\ltimes M$ is equivalent to
$EC_{m}\times_{C_{m}}BM$. On the simplicial level, this equivalence is given
by

$\displaystyle\beta:B(C_{m}\ltimes M)$
$\displaystyle\overset{\sim}{\longrightarrow}EC_{m}\times_{C_{m}}BM,$
$\displaystyle\big{(}(s^{i_{1}},m_{1}),\dots,(s^{i_{q}},m_{q})\big{)}$
$\displaystyle\longmapsto\big{[}(e,s^{i_{1}},\dots,s^{i_{q}}),(s^{i_{1}}\cdot
m_{1},s^{i_{1}+i_{2}}\cdot m_{2},\dots,s^{i_{1}+\dots+i_{q}}\cdot
m_{q})\big{]}.$

Now, for simplicity, we may assume that $\Omega(-)$ stands for the Moore loop
space, so that $C_{m}\ltimes\Omega X$ is strictly associative (a Moore loop is
a pair $(\gamma,t)$ where $t\geq 0$ and $\gamma:[0,t]\to Y$ is a map with
$\gamma(0)=\gamma(t)=*$; multiplication of Moore loops is given by
concatenation of loops and addition of its lengths). Also recall that there is
an identification $\Delta^{q}\cong\\{\mathbf{v}=(v_{1},\dots,v_{q}):0\leq
v_{1}\leq\dots\leq v_{q}\leq 1\\}$; under this identification, the self-
isomorphism $\Phi_{q}:\Delta^{q}\cong\Delta^{q}$ in the proof of Lemma 5.5
becomes the rule that sends a partition $\mathbf{v}=(0\leq v_{1}\leq\dots\leq
v_{q}\leq 1)$ to $1-\mathbf{v}:=(0\leq 1-v_{q}\leq\dots\leq 1-v_{1}\leq 1)$.
There is a $C_{m}$-equivariant map

$\xi:B\Omega X\longrightarrow
X,\quad\big{[}(\gamma_{1},t_{1}),\dots,(\gamma_{q},t_{q}),\mathbf{v}\big{]}\longmapsto\gamma_{q}\cdot\gamma_{q-1}\cdot\dotso\cdot\gamma_{1}\left(\sum_{i=1}^{q}t_{i}v_{i}\right)$

that is an equivalence if $X$ is connected. This map satisfies the property
that

$\xi\left(\big{[}(\overline{\gamma}_{1},t_{1}),\dots,(\overline{\gamma}_{q},t_{q}),\mathbf{v}\big{]}\right)=\xi\left(\big{[}(\gamma_{q},t_{q}),\dots,(\gamma_{1},t_{1}),1-\mathbf{v}\big{]}\right).$

With all of this in mind, one verifies that the following diagram commutes up
to homotopy

${B((C_{m}\ltimes\Omega X)^{\mathrm{op}})}$${B(C_{m}\ltimes\Omega
X)}$${EC_{m}\times_{C_{m}}B\Omega X}$${B(C_{m}\ltimes\Omega
X)}$${EC_{m}\times_{C_{m}}B\Omega
X}$${EC_{m}\times_{C_{m}}X.}$$\scriptstyle{B(\mathrm{inv})}$$\scriptstyle{\iota}$$\scriptstyle{\sim}$$\scriptstyle{\beta}$$\scriptstyle{\sim}$$\scriptstyle{EC_{m}\times_{C_{m}}\xi}$

$\scriptstyle\sim$

$\scriptstyle{\beta}$$\scriptstyle{\sim}$$\scriptstyle{EC_{m}\times_{C_{m}}\xi}$$\scriptstyle{\sim}$

∎

###### Proof of Proposition 5.19.

In the notation of the previous section, we let $X=\Sigma^{\sigma}Y$ now, so
that $G=GX=\Omega^{\sigma}\Sigma^{\sigma}Y$ as a monoid with anti-involution
(i.e. ”inv” now means inversion in the monoid $G$ followed by the
$C_{2}$-action on $X=\Sigma^{\sigma}Y$). For each $m\geq 1$, let us write
$\varrho:GL_{m}(Q_{+}G)\simeq\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])$ for
$u^{-1}$ (meaning $u$ as a wrong way equivalence); it should be thought of as
given by the rule that sends a matrix
$h=(h_{ij})\in\operatorname{Map}(\mathbb{S},\mathbb{S}[G])^{m\times
m}\cong\operatorname{Map}_{G}(\mathbb{S}[G],\mathbb{S}[G])^{m\times m}$ to

${\varrho(h):\bigoplus_{j=1}^{m}\mathbb{S}[G]}$${\bigoplus_{i=1}^{m}\mathbb{S}[G].}$$\scriptstyle{\bigoplus_{j}\big{(}\bigoplus_{i}h_{ij}\big{)}}$

The map $\theta$ of (5.22) is constructed in several steps in [CCGH87, §1],
each of which we now upgrade to the homotopy $C_{2}$-equivariant setting. As
these homotopy $C_{2}$-actions will get mixed up with strict $C_{m}$-actions,
it will be more convenient and clear, at least throughout the first few steps,
to avoid speaking about ”homotopy equivariance” and rather regard a homotopy
involution as what it is, i.e. a map whose square happens to be homotopic to
the identity.

######  Step 1.

For each $m\geq 2$, consider the $D_{m}$-equivariant map of spaces

$\widetilde{\theta}_{m,1}:Y^{\times m}\longrightarrow
GL_{m}(Q_{+}G),\quad(y_{1},\dots,y_{m})\longmapsto\leavevmode\resizebox{142.26378pt}{}{$\begin{pmatrix}1&y_{1}-1&&&&\\\
&1&y_{2}-1&&&\\\ &&1&\ddots&&\\\ &&&\ddots&&\\\ &&&&1&y_{m-1}-1\\\
y_{m}-1&&&&&1\end{pmatrix}$}$

where, if $D_{m}:=\langle s,r\mid s^{m}=r^{2}=rsrs=e\rangle$, the notation is
as follows:

  * •

A point $y\in Y$ is identified in $G$ with the path $\eta(y):=(t\mapsto
t\wedge y)\in G$, which is itself identified with a point in $\\{1\\}\times
QG\subset QS^{0}\times QG\simeq Q_{+}G$. Then $y-1$ is the corresponding point
in $\\{0\\}\times QG\subset Q_{+}G$. Here the $n$-th component of $QS^{0}$ has
been fixed a basepoint $n\in QS^{0}$.

  * •

The action of $D_{m}$ on $(y_{1},\dots,y_{m})\in Y^{\times m}$ is given by

$s\cdot(y_{1},\dots,y_{m}):=(y_{m},y_{1},\dots,y_{m-1}),\quad
r\cdot(y_{1},\dots,y_{m}):=(y^{*}_{m-1},y^{*}_{m-2},\dots,y_{1}^{*},y_{m}^{*}),$

where $y\mapsto y^{*}$ denotes the $C_{2}$-action on $Y$.

  * •

Let $S,R\in GL_{m}({\scaleobj{0.97}{\mathbb{Z}}})$ be the permutation matrices
that send the $i$-th unit vector $e_{i}$ to $e_{i+1}$ and $e_{m+1-i}$ (with
subindexes taken modulo $m$), respectively. Then $D_{m}$ acts on $A\in
GL_{m}(Q_{+}G)$ by

$s\cdot A:=SAS^{-1},\qquad r\cdot A:=RA^{\dagger}R.$

In other words, $r$ acts by transposition along the ”$x=y$”-axis together with
conjugation on $G$.

We also define $\widetilde{\theta}_{1,1}:Y\to
GL_{1}(Q_{+}G)=(Q_{+}G)^{\times}$ by sending $y\in Y$ to $y\in\\{1\\}\times
QG\subset Q_{+}G$. We note that $\widetilde{\theta}_{m,1}(y_{1},\dots,y_{m})$
is homotopy invertible by choosing a path from each of the $y_{i}$’s to the
basepoint $*\in Y$. Then for $m\geq 1$, define $\theta_{m,1}$ as the
composition

${\theta_{m,1}:Y^{\times
m}}$${GL_{m}(Q_{+}G)}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G]).}$$\scriptstyle{\widetilde{\theta}_{m,1}}$$\scriptstyle{\varrho}$$\scriptstyle{\sim}$

By construction, $\theta_{m,1}$ is a $C_{m}$-map as both
$\widetilde{\theta}_{m,1}$ and $u$ are. Recall that $s\in C_{m}\subset D_{m}$
acts on $\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])$ by conjugation with
$S\in\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])$, which we denote by $S_{\\#}$.

######  Step 2.

Recall that the free $\mathbb{E}_{1}$-algebra on a based, connected space $X$
is naturally equivalent to $\Omega\Sigma X$. Therefore, we can extend
$\theta_{m,1}$ to a $C_{m}$-equivariant $\mathbb{E}_{1}$-map

$\theta_{m,2}:\Omega\Sigma(Y^{\times
m})\longrightarrow\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G]).$

For any based space $X$, let us write $\sigma:\Omega X\to(\Omega
X)^{\mathrm{op}}$ for inversion in $\Omega X$ (i.e. reversing the loop
direction). Given a $C_{m}$-space $X$, we will denote
$X^{\mathrm{op}_{C_{m}}}$ for $X$ with the opposite $C_{m}$-action (i.e. that
in which $s$ acts by $s^{-1}$, which is a valid left action as $C_{m}$ is
abelian). If $X$ additionally has an $\mathbb{E}_{1}$-structure, we will write
$X^{\mathrm{op},\mathrm{op}_{C_{m}}}$ for $X$ with both the opposite
$\mathbb{E}_{1}$-structure and the opposite $C_{m}$-action. Then the square of
$\mathbb{E}_{1}$-maps

(5.23) ${\Omega\Sigma(Y^{\times
m})}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${\Omega\Sigma(Y^{\times
m})^{\mathrm{op},\mathrm{op}_{C_{m}}}}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])^{\mathrm{op},\mathrm{op}_{C_{m}}}}$$\scriptstyle{\sigma\circ
r}$$\scriptstyle{\theta_{m,2}}$$\scriptstyle{R_{\\#}\circ\tau_{\epsilon}}$$\scriptstyle{\theta_{m,2}^{\mathrm{op},\mathrm{op}_{C_{m}}}}$

commutes in the homotopy category of $\mathbb{E}_{1}$-spaces with a
$C_{m}$-action. Here $r:\Omega\Sigma(Y^{\times m})\to\Omega\Sigma(Y^{\times
m})^{\mathrm{op}_{C_{m}}}$ is induced by the action of $r\in D_{m}$ on
$Y^{\times m}$ together with the flip of the suspension coordinate, and
$\tau_{\epsilon}$ really stands for (5.19). To see this, consider the diagram

(5.24) ${Y^{\times
m}}$${GL_{m}(Q_{+}G)}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${(Y^{\times
m})^{\mathrm{op}_{C_{m}}}}$${GL_{m}(Q_{+}G)^{\mathrm{op}_{C_{m}}}}$${\Omega\Sigma(Y^{\times
m})^{\mathrm{op}_{C_{m}}}}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])^{\mathrm{op}_{C_{m}}}}$$\scriptstyle{r}$$\scriptstyle{\sigma\circ
r\circ\eta}$$\scriptstyle{\widetilde{\theta}_{m,1}}$$\scriptstyle{r=R_{\\#}\circ\dagger}$$\scriptstyle{\varrho}$$\scriptstyle{\sim}$$\scriptstyle{R_{\\#}\circ\tau_{\epsilon}}$$\scriptstyle{\widetilde{\theta}_{m,1}^{\hskip
2.0pt\mathrm{op}_{C_{m}}}}$$\scriptstyle{\eta^{\mathrm{op}_{C_{m}}}}$$\scriptstyle{\varrho^{\mathrm{op}_{C_{m}}}}$$\scriptstyle{\sim}$$\scriptstyle{\theta_{m,2}^{\mathrm{op}_{C_{m}}}}$

of $C_{m}$-spaces. By definition, $\theta_{m,2}$ is the $\mathbb{E}_{1}$-map
induced from the top horizontal composition in (5.24). Thus, in order to show
that (5.23) homotopy commutes as $C_{m}$-equivariant $\mathbb{E}_{1}$-maps, it
suffices to show that the outer square of (5.24) commutes in the homotopy
category of $C_{m}$-spaces. But each of its subsquares/triangles commute in
this category: indeed the lower subsquare does so by definition of
$\theta_{m,2}$ (after applying $(-)^{\mathrm{op}_{C_{m}}}$), the left
subtriangle and the upper subsquare too by an easy check, and the right
subsquare by Proposition 5.16 and the observation that $R_{\\#}\circ
u=u^{\mathrm{op}_{C_{m}}}\circ R_{\\#}$.

######  Step 3.

The $C_{m}$-equivariant $\mathbb{E}_{1}$-map $\theta_{m,2}$ gives rise to an
$\mathbb{E}_{1}$-map

${\theta_{m,3}:C_{m}\ltimes\Omega\Sigma(Y^{\times
m})}$${C_{m}\ltimes\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G]),}$$\scriptstyle{C_{m}\ltimes\theta_{m,2}}$$\scriptstyle{\mu}$

where $\mu(s^{i},h):=S^{i}h$. Observe that $\mu$ is indeed an
$\mathbb{E}_{1}$-map as

$\mu((s^{i},h)\cdot(s^{j},h^{\prime}))=\mu(s^{i+j},S^{-j}hS^{j}h^{\prime})=S^{i}hS^{j}h^{\prime}=\mu(s^{i},h)\mu(s^{j},h^{\prime}).$

Now from the homotopy commutativity of (5.23), it immediately follows that the
left subsquare in

(5.25) ${C_{m}\ltimes\Omega\Sigma(Y^{\times
m})}$${C_{m}\ltimes\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${C_{m}\ltimes\Omega\Sigma(Y^{\times
m})^{\mathrm{op},\mathrm{op}_{C_{m}}}}$${C_{m}\ltimes\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])^{\mathrm{op},\mathrm{op}_{C_{m}}}}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])^{\mathrm{op}}}$$\scriptstyle{C_{m}\ltimes(\sigma\circ
r)}$$\scriptstyle{\theta_{m,3}}$$\scriptstyle{C_{m}\ltimes\theta_{m,2}}$$\scriptstyle{C_{m}\ltimes(R_{\\#}\circ\tau_{\epsilon})}$$\scriptstyle{\mu}$$\scriptstyle{R_{\\#}\circ\tau_{\epsilon}}$$\scriptstyle{C_{m}\ltimes\theta_{m,2}^{\mathrm{op},\mathrm{op}_{C_{m}}}}$$\scriptstyle{\mu^{\mathrm{op}}}$

commutes in the homotopy category of $\mathbb{E}_{1}$-spaces. Here
$\mu^{\mathrm{op}}(s^{i},h):=hS^{i}$, and since
$\tau_{\epsilon}(S)=S^{\dagger}=S^{-1}$ and $RS=S^{-1}R$, it easily follows
that the right subsquare also commutes as $\mathbb{E}_{1}$-maps. So the outer
square of (5.25) commutes in the homotopy category of $\mathbb{E}_{1}$-spaces.

But given an $\mathbb{E}_{1}$-space $X$ equipped with a $C_{m}$-action, there
is an isomorphism of $\mathbb{E}_{1}$-spaces

$\alpha:C_{m}\ltimes
X^{\mathrm{op},\mathrm{op}_{C_{m}}}\overset{\cong}{\longrightarrow}(C_{m}\ltimes
X)^{\mathrm{op}},\quad(s^{i},x)\longmapsto(s^{i},s^{-i}\cdot x).$

Under this identification, the lower horizontal composition of (5.25) becomes
$\theta_{m,3}^{\mathrm{op}}$, and hence

(5.26) ${C_{m}\ltimes\Omega\Sigma(Y^{\times
m})}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${(C_{m}\ltimes\Omega\Sigma(Y^{\times
m}))^{\mathrm{op}}}$${\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])^{\mathrm{op}}}$$\scriptstyle{\alpha\circ(C_{m}\ltimes(\sigma\circ
r))}$$\scriptstyle{\theta_{m,3}}$$\scriptstyle{R_{\\#}\circ\tau_{\epsilon}}$$\scriptstyle{\theta_{m,3}^{\mathrm{op}}}$

is commutative in the homotopy category of $\mathbb{E}_{1}$-spaces.

######  Step 4.

We wish to deloop (5.26), viewing the vertical maps as anti-involutions of
their respective domains, and appealing to Lemma 5.5 to do so. But by Lemma
5.21, the delooping of the anti-involution
$\alpha\circ(C_{m}\ltimes(\sigma\circ r))$ is homotopic to the delooping of
the involution $\mathrm{inv}\circ\alpha\circ(C_{m}\ltimes(\sigma\circ r))$,
where $\mathrm{inv}$ stands for inversion in the $\mathbb{E}_{1}$-space
$C_{m}\ltimes\Omega\Sigma(Y^{\times m})$. It is given explictly by
$\mathrm{inv}(s^{i},\gamma):=(s^{-i},s^{i}\cdot\sigma(\gamma))$, and hence we
see that

$\mathrm{inv}\circ\alpha\circ(C_{m}\ltimes(\sigma\circ
r)):(s^{i},\gamma)\longmapsto(s^{-i},r\cdot\gamma).$

We denote this map simply by $\mathrm{inv}\ltimes r$. From now on we treat $r$
as the action map on the $D_{m}$-space $\Sigma^{\sigma}(Y^{\times m})$, where
$\sigma$ is seen as a $D_{m}$-representation on which $C_{m}$ acts trivially.
Putting this together, the delooped version of (5.26) yields a homotopy
commutative square of spaces

${B(C_{m}\ltimes\Omega\Sigma^{\sigma}(Y^{\times
m}))}$${B\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${B(C_{m}\ltimes\Omega\Sigma^{\sigma}(Y^{\times
m}))}$${B\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G]),}$$\scriptstyle{B(\mathrm{inv}\ltimes
r)}$$\scriptstyle{B(\theta_{m,3})}$$\scriptstyle{\overline{B}(R_{\\#}\circ\tau_{\epsilon})}$$\scriptstyle{B(\theta_{m,3})}$

where the notation $\overline{B}(-)$ stands for delooping in the sense of
Lemma 5.5.

To simplify the terms in this last diagram, first observe that as $Y^{\times
m}$ is connected, we have

$B(C_{m}\ltimes\Omega\Sigma^{\sigma}(Y^{\times m}))\simeq
ED_{m}\times_{C_{m}}B\Omega\Sigma^{\sigma}(Y^{\times m})\simeq
ED_{m}\times_{C_{m}}\Sigma^{\sigma}(Y^{\times m}).$

The inversion on $C_{m}$ coincides with the residual
$C_{2}=D_{m}/C_{m}$-action on $C_{m}$ by conjugation, which explains why we
chose to write $ED_{m}$ instead of $EC_{m}$. As for the right hand side, note
that $R_{\\#}$ is an inner automorphism of
$\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])$, and hence it induces a map
homotopic to the identity on the classifying space level [AM04, §II, Thm.
1.9]. But delooping is functorial, so
$\overline{B}(R_{\\#}\circ\tau_{\epsilon})$ and
$\overline{B}(\tau_{\epsilon})=:\tau_{\epsilon}$ are homotopic involutions on
$B\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])$. All together, we obtain a
homotopy $C_{2}$-equivariant map

$\theta_{m,4}:ED_{m}\times_{C_{m}}\Sigma^{\sigma}(Y^{\times
m})\xlongrightarrow{B(\theta_{m,3})^{+}}B\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])^{+}\subset{\scaleobj{0.97}{\mathbb{Z}}}\times
B\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])^{+}\to
A(\Sigma^{\sigma}Y;\epsilon),$

where the last map is the passage to the colimit as $m\to\infty$ (see (5.18)).

######  Step 5.

The following diagram commutes up to homotopy:

${\theta_{m,4}:ED_{m}/C_{m}\simeq
BC_{m}}$${B\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S})}$${A(*;\epsilon)}$${\theta_{m,4}:ED_{m}\times_{C_{m}}\Sigma^{\sigma}(Y^{\times
m})}$${B\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S}[G])}$${A(\Sigma^{\sigma}Y;\epsilon),}$$\scriptstyle{Bj}$

where $j:C_{m}\to\mathrm{Aut}_{G}(\oplus^{m}\mathbb{S})$ is the inclusion of
the permutation automorphisms. As
$A(-;\epsilon)=\Omega^{\infty}\mathbf{A}(-;\epsilon)$, we can adjoin the
$\Omega^{\infty}(-)$ to get a similar homotopy commutative diagram of spectra.
Then passing to vertical cofibres and noting that $C_{m}$ acts trivially on
the suspension coordinate of $\Sigma^{\sigma}(Y^{\times m})$, we get a
homotopy $C_{2}$-equivariant map of spectra

$\theta_{m,5}:\Sigma^{\infty+\sigma}\big{(}(ED_{m})_{+}\wedge_{C_{m}}Y^{\times
m}\big{)}\longrightarrow\widetilde{\mathbf{A}}(\Sigma^{\sigma}Y;\epsilon).$

######  Step 6.

Now by [CCGH87, Lem. 1.4] (see also [CC87, Lem. 2.4]), the obvious projection
$(ED_{m})_{+}\wedge_{C_{m}}Y^{\times m}\to(ED_{m})_{+}\wedge_{C_{m}}Y^{\wedge
m}$ has a stable section
$\Sigma^{\infty}\big{(}(ED_{m})_{+}\wedge_{C_{m}}Y^{\wedge
m}\big{)}\to\Sigma^{\infty}\big{(}(ED_{m})_{+}\wedge_{C_{m}}Y^{\times
m}\big{)}$ that is $D_{m}/C_{m}$-equivariant. This observation gives rise to a
homotopy $C_{2}$-equivariant map of spectra

$\boldsymbol{\theta}_{m,6}:\Sigma^{\infty+\sigma}\big{(}(ED_{m})_{+}\wedge_{C_{m}}Y^{\wedge
m}\big{)}\longrightarrow\widetilde{\mathbf{A}}(\Sigma^{\sigma}Y;\epsilon).$

Finally set $\boldsymbol{\theta}$ to be

${\boldsymbol{\theta}:\bigvee_{m\geq
1}\Sigma^{\infty+\sigma}\big{(}(ED_{m})_{+}\wedge_{C_{m}}Y^{\wedge
m}\big{)}}$${\bigvee_{m\geq
1}\widetilde{\mathbf{A}}(\Sigma^{\sigma}Y;\epsilon)}$${\widetilde{\mathbf{A}}(\Sigma^{\sigma}Y;\epsilon).}$$\scriptstyle{\bigvee_{m\scaleobj{0.85}{\geq}1}(\boldsymbol{\theta}_{m,6})}$

This map is homotopy $C_{2}$-equivariant by construction, and non-
equivariantly yields (5.22) after applying $\Omega^{\infty+\sigma}(-)$. This
latter map is an equivalence of infinite loop spaces by [CCGH87, Thm. 1.6],
and as both the domain and codomain of $\boldsymbol{\theta}$ are
$1$-connective, it follows that $\boldsymbol{\theta}$ is itself an equivalence
of spectra. This concludes the proof of Proposition 5.19. ∎

######  Corollary 5.22.

Let $Y$ be a connected, based $C_{2}$-space. There is an equivalence of
spectra

$\boldsymbol{\theta}:\bigvee_{m\geq
2}\Sigma^{\infty+\sigma}\big{(}(ED_{m})_{+}\wedge_{C_{m}}Y^{\wedge
m}\big{)}\overset{\sim}{\longrightarrow}\widetilde{\mathbf{Wh}}{\vphantom{\mathrm{Wh}}}^{\mathrm{Diff}}(\Sigma^{\sigma}Y;\epsilon)$

that is $C_{2}$-equivariant up to homotopy. Here
$\widetilde{\mathbf{Wh}}{\vphantom{\mathrm{Wh}}}^{\mathrm{Diff}}(-):=\operatorname{hofib}(\mathbf{Wh}^{\mathrm{Diff}}(-)\to\mathbf{Wh}^{\mathrm{Diff}}(*))$.

###### Proof.

It is clear from the construction that the map

$\Sigma^{\infty}(\Sigma^{\sigma}Y)\simeq\Sigma^{\infty+\sigma}\big{(}(ED_{1})_{+}\wedge_{C_{1}}Y^{\wedge
1}\big{)}\xlongrightarrow{\boldsymbol{\theta}_{1,6}}\widetilde{\mathbf{A}}(\Sigma^{\sigma}Y;\epsilon)$

is the (reduced version of the) usual inclusion of the stable homotopy into
$A$-theory. Thus its cofibre is
$\widetilde{\mathbf{Wh}}{\vphantom{\mathrm{Wh}}}^{\mathrm{Diff}}(\Sigma^{\sigma}Y;\epsilon)$,
and the claim follows immediately. ∎

######  Remark 5.23.

As the reader may have noticed by now, the last two sections are a tiny bit
technical, and one may wonder if there could be alternative approaches to deal
with them. Such an approach that may come to mind is to use trace methods to
analyse $\widetilde{A}(\Sigma^{\sigma}Y;\epsilon)$, since it coincides (non-
equivariantly) with the reduced $TC$ of $\mathbb{S}[\Omega\Sigma Y]$ (as $Y$
is connected). In fact, recent developments have been made towards a
$C_{2}$-equivariant version of topological cyclic homology for ring spectra
with anti-involutions, commonly known as real topological cyclic homology (cf.
[Hø16, HM16, DMP21]). This approach has two caveats:

  * •

A real cyclotomic trace map does not yet exist (at the time of writing). The
construction of such a map was supposed to appear in [HM16], but it never saw
the light in the end. This is, nevertheless, current work in progress by
Harpaz–Nikolaus–Shah [HNS21, p. 24].

  * •

Even though much is known about the $p$-complete homotopy type of the $TC$ of
spherical group rings (cf. [BCC+96] or [NS18, §4.3]), the analysis of its
integral homotopy type does not seem to be present in the literature.

For these two reasons, we prefer to proceed as we have.

##  6\. The homotopy type of spaces of long knots

This section is devoted to Theorem B, which describes the homotopy type of
$\operatorname{Emb}_{\partial}(D^{p},D^{d})$ for $p\leq d-3$ and $d\geq 5$,
localised at odd primes and up to the concordance embedding stable range
$\phi_{C\mathrm{Emb}}(d,p)$. After its proof, which will not take too much
effort given the results in the preceeding sections, we will draw some
conclusions on the homotopy groups of spaces of long knots. For convenience
let us recall the statement of Theorem B. Recall that $\psi_{m}$ stands for
the real $m$-dimensional representation of the dihedral group $D_{m}$ and
$\sigma$ for the sign representation, regarded in this statement as a
$D_{m}$-representation by restricting along the determinant
$D_{m}\xhookrightarrow{}O(2)\overset{\det}{\to}\\{\pm 1\\}=C_{2}$.

######  Theorem (Theorem B).

For $p\leq d-3$ and $d\geq 5$, there exists a homotopy fibre sequence which,
after localising away from $2$ and taking $(\phi_{C\mathrm{Emb}}(d,p)-1)$-th
Postnikov sections, takes the form

${\prod_{m\scaleobj{0.85}{\geq}2}\Omega^{\infty}\big{(}\big{(}\mathbb{S}^{(d+1)\cdot(\sigma-1)}\wedge\mathrm{Th}(\psi_{m}\otimes(d-p-3+\sigma)\mid_{C_{m}})\big{)}_{hC_{2}}\big{)}}$${\operatorname{Emb}_{\partial}(D^{p},D^{d})}$${\Omega^{p}\operatorname{hofib}\big{(}G(d-p)/O(d-p)\to
G/O\big{)}.}$

It is split for $p\geq 2$, and splits after being looped once for $p=1$.

Recall from Remark 1.4($ii$) that the space
$\operatorname{Emb}_{\partial}(D^{p},D^{d})$ (and
$\widetilde{\operatorname{Emb}}_{\partial}(D^{p},D^{d})$ too) can be
localised.

###  6.1. Proof of Theorem B

Recall from (1.4) that we write
$\operatorname{Emb}_{\partial}^{(\sim)}(P,M):=\operatorname{hofib}_{\iota}(\operatorname{Emb}_{\partial}(P,M)\to\widetilde{\operatorname{Emb}}_{\partial}(P,M))$
when $\iota$ is clear from the context. We saw in Corollary 4.3 that the
fibration sequence

(6.1)
${\operatorname{Emb}^{(\sim)}_{\partial}(D^{p},D^{d})}$${\operatorname{Emb}_{\partial}(D^{p},D^{d})}$${\widetilde{\operatorname{Emb}}_{\partial}(D^{p},D^{d}),}$

upon localising at odd primes and taking $(\phi_{C\mathrm{Emb}}(d,p)-1)$-th
Postnikov sections, is split for $2\leq p\leq d-3$, and splits for $p=1$ after
looping once. So we need to describe the exterior terms of (6.1), after
inverting $2$.

For the block embeddings, the graphing map

(6.2)
$\Gamma:\Omega^{p}\widetilde{\operatorname{Emb}}(*,D^{d-p})\overset{\sim}{\longrightarrow}\widetilde{\operatorname{Emb}}_{\partial}(D^{p},D^{d-p}\times
D^{p})\cong\widetilde{\operatorname{Emb}}_{\partial}(D^{p},D^{d})$

of (4.1) is an equivalence by inspection. Then by [GKW01, Thm. 2.2.1] and the
example right after it, when $d-p\geq 3$ and $d\geq 5$ (see Remark 6.1 below),
it follows that

(6.3)
$\widetilde{\operatorname{Emb}}_{\partial}(D^{p},D^{d})\simeq\Omega^{p}\operatorname{hofib}(O/O(d-p)\to
G/G(d-p))\simeq\Omega^{p}\operatorname{hofib}(G(d-p)/O(d-p)\to G/O),$

yielding the base of (6.1).

######  Remark 6.1.

The equivalence (6.3) is only valid if $d-p\geq 3$ and $d\geq 5$. As pointed
out right after [GKW01, Thm. 2.2.1], the second condition is not that
important. For instance in the case $p=1$ and $d=4$, it follows directly from
(6.2) and (6.3) that

$\Omega\widetilde{\operatorname{Emb}}_{\partial}(D^{1},D^{4})\simeq\widetilde{\operatorname{Emb}}_{\partial}(D^{2},D^{5})\simeq\Omega^{2}\operatorname{hofib}(G(3)/O(3)\to
G/O).$

The codimension condition $d-p\geq 3$, however, is essential.

For the fibre of (6.1), we know by Theorem A that, for
$N=\phi_{C\mathrm{Emb}}(d,p)-1$, there is an equivalence

$\tau_{\scaleobj{0.85}{\leq}N}\operatorname{Emb}^{(\sim)}_{\partial}(D^{p},D^{d})\simeq\tau_{\scaleobj{0.85}{\leq}N}\
\Omega^{\infty}\left(\Sigma^{-1}\Theta(D^{d},S^{d-p-1}\times
D^{p+1})_{hC_{2}}\right).$

We now use Corollaries 5.15 and 5.22 to describe the right hand side of the
equivalence above.

######  Proposition 6.2.

There is an equivalence

$\Omega^{\infty}\left(\Sigma^{-1}\Theta(D^{d},S^{d-p-1}\times
D^{p+1})_{hC_{2}}\right)\simeq_{[\frac{1}{2}]}\prod_{m\scaleobj{0.85}{\geq}2}\Omega^{\infty}\big{(}\big{(}\mathbb{S}^{(d+1)\cdot(\sigma-1)}\wedge\mathrm{Th}(\psi_{m}\otimes(d-p-3+\sigma)\mid_{C_{m}})\big{)}_{hC_{2}}\big{)}.$

###### Proof.

First observe that there is a homotopy $C_{2}$-equivariant equivalence
$S^{2}\approx S^{2\sigma}$. So by Corollary 5.22, for each $n\geq 2$ there is
a homotopy $C_{2}$-equivariant equivalence of spectra

$\widetilde{\mathbf{Wh}}{\vphantom{\mathrm{Wh}}}^{\mathrm{Diff}}(S^{n};\epsilon)\approx\widetilde{\mathbf{Wh}}{\vphantom{\mathrm{Wh}}}^{\mathrm{Diff}}(\Sigma^{\sigma}S^{n-2+\sigma};\epsilon)\approx\bigvee_{m\geq
2}\Sigma^{\infty+\sigma}\big{(}S^{\psi_{m}\otimes(n-2+\sigma)}_{hC_{m}}\big{)}=\bigvee_{m\geq
2}\mathbb{S}^{\sigma}\wedge\mathrm{Th}(\psi_{m}\otimes(n-2+\sigma)\mid_{C_{m}}).$

Using this for $n=d-p-1\geq 2$, we obtain a chain of equivalences

$\displaystyle\Omega^{\infty}\big{(}\Sigma^{-1}\Theta(D^{d},S^{d-p-1}\times
D^{p+1})_{hC_{2}}\big{)}$
$\displaystyle\simeq_{[\frac{1}{2}]}\Omega^{\infty}\left((S^{d\cdot(\sigma-1)-1}\wedge\widetilde{\mathbf{Wh}}{\vphantom{\mathrm{Wh}}}^{\mathrm{Diff}}(S^{d-p-1}))_{hC_{2}}\right)$
$\displaystyle\simeq_{[\frac{1}{2}]}\Omega^{\infty}\big{(}\big{(}\bigvee_{m\geq
2}\mathbb{S}^{(d+1)\cdot(\sigma-1)}\wedge\mathrm{Th}(\psi_{m}\otimes(d-p-3+\sigma)\mid_{C_{m}})\big{)}_{hC_{2}}\big{)}.$
$\displaystyle=\prod_{m\scaleobj{0.85}{\geq}2}\Omega^{\infty}\big{(}\big{(}\mathbb{S}^{(d+1)\cdot(\sigma-1)}\wedge\mathrm{Th}(\psi_{m}\otimes(d-p-3+\sigma)\mid_{C_{m}})\big{)}_{hC_{2}}\big{)}.$

The first equivalence follows from the observation that
$\widetilde{\mathbf{Wh}}{\vphantom{\mathrm{Wh}}}^{\mathrm{Diff}}(S^{d-p-1})\simeq\Sigma^{-1}\mathbf{Wh}^{\mathrm{Diff}}(D^{p},S^{d-p-1};\epsilon)$
because both $\tau_{\epsilon}$ and $\mathbf{Wh}^{\mathrm{Diff}}(-)$ are
homotopy invariants of $(-)$, together with Corollary 5.15. The second
equivalence is a consequence of the previous argument and Proposition 5.3.
This establishes the desired equivalence. ∎

All together, this completes the proof of Theorem B. ∎

######  Remark 6.3 (Topological version of Theorem B).

The space $\operatorname{Emb}_{\partial}^{\mathrm{Top}}(D^{p},D^{d})$ of
topological long knots is contractible (for all $p\leq d$) by the Alexander
trick. We could still be interested in the homotopy type of the space
$\operatorname{Emb}_{\partial_{0}}^{\mathrm{Top}}(D^{p}\times D^{d-p},D^{d})$
of thickened topological long knots with $p\leq d-3$, and one can get a
description of it localised away from $2$ and up to the concordance embedding
stable range, similar to the one in Theorem B—let us explain how. As before,
we have a homotopy fibre sequence

${\operatorname{Emb}_{\partial_{0}}^{\mathrm{Top},(\sim)}(D^{p}\times
D^{d-p},D^{d})}$${\operatorname{Emb}_{\partial_{0}}^{\mathrm{Top}}(D^{p}\times
D^{d-p},D^{d})}$${\widetilde{\operatorname{Emb}}_{\partial_{0}}^{\mathrm{Top}}(D^{p}\times
D^{d-p},D^{d})}$

which, upon localising at odd primes and taking
$(\phi_{C\mathrm{Emb}}(d,p)-1)$-th Postnikov sections, is split for $2\leq
p\leq d-3$, and splits for $p=1$ after looping once. So we should describe the
side terms.

For the block embeddings, consider the space

$B\widetilde{\mathrm{Top}}(q):=\operatorname{holim}\left(\leavevmode\hbox
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)$

which is responsible for the classification of topological block normal
bundles if $q\geq 3$ (cf. [RS68b, §2], [RS68a] and [Wal99, §11]). As
$\widetilde{\operatorname{Emb}}^{\mathrm{Top}}_{\partial}(D^{p},D^{d})$ is
contractible by the Alexander trick, it then follows that

$\widetilde{\operatorname{Emb}}^{\mathrm{Top}}_{\partial_{0}}(D^{p}\times
D^{d-p},D^{d})\simeq\operatorname{Map}_{\partial}(D^{p},\widetilde{\mathrm{Top}}(d-p))=\Omega^{p}\widetilde{\mathrm{Top}}(d-p).$

As for the pseudoisotopy embeddings, the topological version of Theorem A (see
Remark 1.3) tells us that for $N=\phi_{C\mathrm{Emb}(d,p)}-1$, there is an
equivalence

(6.4)
$\tau_{\scaleobj{0.85}{\leq}N}\operatorname{Emb}_{\partial_{0}}^{\mathrm{Top},(\sim)}(D^{p}\times
D^{d-p},D^{d}))\simeq\tau_{\scaleobj{0.85}{\leq}N}\
\Omega^{\infty}\left(\Sigma^{-1}\Theta^{\mathrm{Top}}(D^{d},S^{d-p-1}\times
D^{p+1})_{hC_{2}}\right),$

where $\Theta^{\mathrm{Top}}(M)$ stands for the first orthogonal derivative of
the functor $B\mathrm{Homeo}^{b}_{\partial}(M\times-)$ (as for
$\Theta(M)=\Theta^{\mathrm{Diff}}(M)$, this is a $C_{2}$-spectrum whose
underlying spectrum is (equivalent to)
$\Sigma^{-1}\mathbf{Wh}^{\mathrm{Top}}(M)$). Noting that there is fibre
sequence of $C_{2}$-spectra
$\mathbf{Wh}^{\mathrm{Diff}}(M;\epsilon)\to\mathbf{Wh}^{\mathrm{Top}}(M;\epsilon)\to\Sigma\mathbf{Wh}^{\mathrm{Diff}}(*;\epsilon)\wedge
M_{+}$, one verifies that the infinite loop space in the right hand side of
(6.4) fits in a fibre sequence away from $2$

${\prod_{m\scaleobj{0.85}{\geq}2}\Omega^{\infty}\big{(}\big{(}\mathbb{S}^{(d+1)\cdot(\sigma-1)}\wedge\mathrm{Th}(\psi_{m}\otimes(d-p-3+\sigma)\mid_{C_{m}})\big{)}_{hC_{2}}\big{)}}$${\Omega^{\infty}\left(\Sigma^{-1}\Theta^{\mathrm{Top}}(D^{d},S^{d-p-1}\times
D^{p+1})_{hC_{2}}\right)}$${\Omega^{\infty}\big{(}(S^{d\cdot\sigma-p-2}\wedge\mathbf{Wh}^{\mathrm{Diff}}(*;\epsilon))_{hC_{2}}\big{)}.}$

In particular, it is easy to check (e.g. rationally) that the left hand side
of (6.4) is not contractible (at least if $d$ is sufficiently large). This was
claimed in Remark 3.4.

###  6.2. On the homotopy groups of spaces of long knots

We can get plenty of information about the homotopy groups of
$\operatorname{Emb}_{\partial}(D^{p},D^{d})$ from Theorem B. First observe
that by Morlet’s lemma of disjunction [BLR06, Thm. 3.1] (and Proposition 3.3
to reduce to the codimension zero case), the pseudoisotopy embedding space
$\operatorname{Emb}_{\partial}^{(\sim)}(D^{p},D^{d})$ is at least
$(2(d-p-2)-1)$-connected. So by (6.3), it follows that

(6.5)
$\pi_{*}(\operatorname{Emb}_{\partial}(D^{p},D^{d}))\cong\pi_{*+p}(\operatorname{hofib}(G(d-p)/O(d-p)\to
G/O)),\quad*<2(d-p-2).$

######  Remark 6.4.

This should be compared to work of Budney [Bud08, Prop. 3.9]. The main result
there is the computation of the first non-trivial homotopy group of
$\operatorname{Emb}_{\partial}(D^{p},D^{d})$ for $d-p\geq 3$, which lies in
degree $2d-3p-3$, together with a geometric interpretation of the generators.
From our point of view, he shows that $\operatorname{hofib}(G(d-p)/O(d-p)\to
G/O)$ is exactly $(2d-2p-4)$-connected, which follows by work of Haefliger
(see Section 3, Equation 4.11 and Corollary 6.6 of [Hae66]), and computes
$\pi_{2d-2p-3}(\operatorname{hofib}(G(d-p)/O(d-p)\to G/O))$. Moreover, it is
stated in [Bud08, Prop. 3.9(1)] that the graphing map

$\Gamma:\pi_{*}(\Omega\operatorname{Emb}_{\partial}(D^{p-1},D^{d-1}))\longrightarrow\pi_{*}(\operatorname{Emb}_{\partial}(D^{p},D^{d}))$

is surjective for $*\leq 2d-2p-5$. From (6.3) and the fact that
$\operatorname{Emb}^{(\sim)}_{\partial}(D^{p},D^{d})$ is
$(2d-2p-5)$-connected, we see that it is in fact an isomorphism.

Recall that $\phi_{C\mathrm{Emb}}(d,p)\geq 2d-p-5$ by work of
Goodwillie–Krannich–Kupers [GKK23], and so the pseudoisotopy embedding space
$\operatorname{Emb}^{(\sim)}_{\partial}(D^{p},D^{d})$ has interesting homotopy
in degrees from $2d-2p-4$ up to that range that we can understand by Theorem
B. Let $E^{m}$ denote the $C_{2}=D_{m}/C_{m}$-spectrum

$E^{m}:=\mathbb{S}^{(d+1)\cdot(\sigma-1)}\wedge\mathrm{Th}(\psi_{m}\otimes(d-p-3+\sigma)\mid_{C_{m}})\simeq\mathbb{S}^{(d+1)\cdot(\sigma-1)}\wedge\mathbb{S}^{\psi_{m}\otimes(d-p-3+\sigma)}_{hC_{m}},$

for $m\geq 2$. Then by Theorem B, for any odd prime $\ell$ there are
isomorphisms in degrees $*\leq\phi_{C\mathrm{Emb}}(d,p)-1$

$\pi_{*}(\operatorname{Emb}_{\partial}(D^{p},D^{d}))_{(\ell)}\cong\pi_{*+p}(\operatorname{hofib}(G(d-p)/O(d-p)\to
G/O))_{(\ell)}\oplus\bigoplus_{m\geq 2}\pi_{*}^{s}(E^{m}_{hC_{2}})_{(\ell)},$

and if $\phi=\phi_{C\mathrm{Emb}}(d,p)$, there is also an exact sequence of
abelian groups

${\bigoplus_{m\geq
2}\pi_{\phi}^{s}(E^{m}_{hC_{2}})_{(\ell)}}$${\pi_{\phi}(\operatorname{Emb}_{\partial}(D^{p},D^{d}))_{(\ell)}}$${\pi_{\phi+p}(\operatorname{hofib}(G(d-p)/O(d-p)\to
G/O))_{(\ell)},}$

where $A_{(\ell)}$ denotes $A\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(\ell)}$,
for $A$ an abelian group. It remains to understand the groups
$\pi_{*}^{s}(E^{m}_{hC_{2}})_{(\ell)}$, which are easier to study when $\ell$
is coprime to $m$.

######  Proposition 6.5.

Let $m\geq 2$ and $d-p\geq 3$. For $\ell\nmid 2m$ a prime,

  * •

if $d$ is even and $p$ is even, then

$\pi_{*}^{s}(E^{m}_{hC_{2}})_{(\ell)}\cong\left\\{\begin{array}[]{cl}\pi_{*-m(d-p-2)}^{s}\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(\ell)},&m=3,5,7,\dots\\\
0,&\text{otherwise}.\end{array}\right.$

  * •

if $d$ is odd and $p$ is odd, then

$\pi_{*}^{s}(E^{m}_{hC_{2}})_{(\ell)}\cong\left\\{\begin{array}[]{cl}\pi_{*-m(d-p-2)}^{s}\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(\ell)},&m=2,4,6,\dots\\\
0,&\text{otherwise}.\end{array}\right.$

  * •

if $d$ is even and $p$ is odd, then

$\pi_{*}^{s}(E^{m}_{hC_{2}})_{(\ell)}\cong\left\\{\begin{array}[]{cl}\pi_{*-m(d-p-2)}^{s}\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(\ell)},&m=5,9,13,\dots\\\
0,&\text{otherwise}.\end{array}\right.$

  * •

if $d$ is odd and $p$ is even, then

$\pi_{*}^{s}(E^{m}_{hC_{2}})_{(\ell)}\cong\left\\{\begin{array}[]{cl}\pi_{*-m(d-p-2)}^{s}\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(\ell)},&m=3,7,11,\dots\\\
0,&\text{otherwise}.\end{array}\right.$

######  Remark 6.6 (Rational homotopy of spaces of long knots).

The rational homology and homotopy of
$\operatorname{Emb}_{\partial}(D^{p},D^{d})$ for $d-p\geq 3$ has been
extensively studied in recent years (see e.g. [Tur10, AT14, AT15]) through the
lens of embedding calculus and its relation to the little disks operads and
their formality, finally culminating in the work of Fresse–Turchin–Willwacher
[FTW17]. There they compute the rational homotopy groups of
$\overline{\operatorname{Emb}}_{\partial}(D^{p},D^{d}):=\operatorname{hofib}_{\iota}(\operatorname{Emb}_{\partial}(D^{p},D^{d})\to\mathrm{Imm}_{\partial}(D^{p},D^{d}))$
as the homology of the hairy graph complex (shifted appropriately).
Observationally, our results correspond to the $0$\- and $1$-loop order parts
of this graph complex up to degree $\phi_{C\mathrm{Emb}}(d,p)\geq 2d-p-5$,
where higher loop orders are still not seen. More precisely, the $0$-loop part
corresponds to the rational homotopy of $G/G(d-p)$, the lowest summand (i.e.
$m=1$ when $d-p$ is even) of the $1$-loop part appears as that of $O/O(d-p)$,
and the higher summands of the $1$-loop part come from the rational homotopy
of the spectra $E^{m}_{hC_{2}}$ for $m\geq 2$, which we just computed. It is
worth noting that:

  * •

The first non-trivial rational homotopy group of
$\operatorname{Emb}_{\partial}(D^{p},D^{d})$ coming from the $2$-loop part of
the hairy graph complex lies in degree $2d-p-4$ when both $d$ and $p$ are odd
(cf. [FTW17, Eq. 3]). Therefore, the lower bound
$\phi_{C\mathrm{Emb}}(d,p)\geq 2d-p-5$ on the concordance stable range of
Goodwillie–Krannich–Kupers is very much sharp.

  * •

The $1$-loop part of the hairy graph complex seems to be completely generated
by the spectra $E^{m}_{hC_{2}}$ for $m\geq 2$ in all degrees outside of the
concordance embedding stable range. In other words, the computations in
[FTW17] give evidence for the existence of a rational left splitting of the
Weiss–Williams map

$\Phi^{\operatorname{Emb}}:\operatorname{Emb}^{(\sim)}_{\partial}(D^{p},D^{d})\longrightarrow\prod_{m\scaleobj{0.85}{\geq}2}\Omega^{\infty}E^{m}_{hC_{2}}.$

To investigate this, one should first understand the attachment of the second
orthogonal derivative $\Theta F^{(2)}$ of the orthogonal functor
$F(U):=\operatorname{Emb}^{b}_{\partial}(D^{p}\times U,D^{d}\times U)$ to its
Taylor tower (2.1). It would also be interesting to understand the integral
picture.

###### Proof of Proposition 6.5.

When $\ell$ is coprime to $2m$, we have that

$\pi_{*}^{s}(E^{m}_{hC_{2}})_{(\ell)}\cong
H_{0}(D_{m};\pi_{*}^{s}(\mathbb{S}^{(d+1)(\sigma-1)+\psi_{m}\otimes(d-p-3+\sigma)}))_{(\ell)}$

by the Bousfield–Kan spectral sequence, because the higher group homology of
$D_{m}$ is $2m$-torsion. Let $t$ and $r$ denote the generators of $D_{m}$ with
$t^{m}=e$ and $rtr=t^{-1}$ such that

$\displaystyle\psi_{m}(t):$ $\displaystyle\
{\scaleobj{0.97}{\mathbb{R}}}^{m}\ni(a_{1},\dots,a_{m})\longmapsto(a_{m},a_{1},\dots,a_{m-1}),$
$\displaystyle\psi_{m}(r):$ $\displaystyle\
{\scaleobj{0.97}{\mathbb{R}}}^{m}\ni(a_{1},\dots,a_{m})\longmapsto(a_{m},a_{m-1},\dots,a_{1}).$

Then $t$ and $r$ act on the group
$\pi^{s}_{*}(\mathbb{S}^{(d+1)(\sigma-1)+\psi_{m}\otimes(d-p-3+\sigma)})$ by
$(-1)^{\epsilon_{t}}$ and $(-1)^{\epsilon_{r}}$, respectively, where

$\displaystyle\epsilon_{t}=(m-1)(d-p-2),\qquad\epsilon_{r}$
$\displaystyle=\underbrace{d+1}_{(1)}+\underbrace{\frac{1}{2}m(m-1)(d-p-3)}_{(2)}+\underbrace{m+\frac{1}{2}m(m-1)}_{(3)}$
$\displaystyle\equiv d+1+m+\frac{1}{2}m(m-1)(d-p-2)\mod 2.$

The terms ($1$), ($2$) and ($3$) are the contributions coming, respectively,
from $(d+1)(\sigma-1)$, $\psi_{m}\otimes(d-p-3)$ and $\psi_{m}\otimes\sigma$.
One then readily verifies that the groups
$H_{0}(D_{m};\pi^{s}_{*}(\mathbb{S}^{(d+1)(\sigma-1)+\psi_{m}\otimes(d-p-3+\sigma)}))$
are given by the fomuli in the statement. ∎

A bit more interesting are the homotopy groups
$\pi^{s}_{*}(E^{m}_{hC_{2}})_{(\ell)}$ when $\ell$ is odd but divides $m$. We
treat the case when $\ell=m=3$, which hopefully serves as a sample computation
for other cases.

######  Proposition 6.7.

The first few homotopy groups
$\pi_{*}^{s}(E^{3}_{hC_{2}})\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(3)}$ of the
spectrum $E^{3}_{hC_{2}}$, localised at $3$ and when $d-p=3$, are given in
Table 1. Equally coloured groups in this table correspond to the same case
depending on whether certain differentials in Figure 5 vanish or not. Entries
containing ”$?$” correspond to potentially more complicated answers that do
not conveniently fit in the table.

Table 1. $\pi_{*}^{s}(E^{3}_{hC_{2}})\otimes\mathbb{Z}_{(3)}$ for $d-p=3$ for low values of $*\geq 3$. $*$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---|---|---|---
$p$ even | ${\scaleobj{0.97}{\mathbb{Z}}}_{(3)}$ | $0$ | $0$ | ${\scaleobj{0.97}{\mathbb{Z}}}/9$ | $0$ | $0$ | $0$ | ${\scaleobj{0.97}{\mathbb{Z}}}/9$ | $0$ | $0$ | ${\scaleobj{0.97}{\mathbb{Z}}}/3$
$p$ odd | $0$ | ${\scaleobj{0.97}{\mathbb{Z}}}/3$ | $0$ | $0$ | $0$ | ${\scaleobj{0.97}{\mathbb{Z}}}/3$ | $0$ | $0$ | ${\scaleobj{0.97}{\mathbb{Z}}}/3$ | ${\scaleobj{0.97}{\mathbb{Z}}}/9$ | $0$
$*$ | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24
$p$ even | ${\scaleobj{0.97}{\mathbb{Z}}}/27$ | $0$ | $0$ | $\begin{array}[]{c}\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{{\scaleobj{0.97}{\mathbb{Z}}}/3}\\\\[3.0pt] \color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}{\scaleobj{0.97}{\mathbb{Z}}}/3\oplus{\scaleobj{0.97}{\mathbb{Z}}}/3\end{array}$ | $\begin{array}[]{c}\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\scaleobj{0.97}{\mathbb{Z}}}/3^{4}\\\\[2.0pt] \color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}{\scaleobj{0.97}{\mathbb{Z}}}/3^{5}\end{array}$ | $0$ | $?$ | $?$ | ${\scaleobj{0.97}{\mathbb{Z}}}/9$ | ${\scaleobj{0.97}{\mathbb{Z}}}/3$ | $0$
$p$ odd | ${\scaleobj{0.97}{\mathbb{Z}}}/3$ | ${\scaleobj{0.97}{\mathbb{Z}}}/3$ | ${\scaleobj{0.97}{\mathbb{Z}}}/3$ | $0$ | $0$ | $0$ | ${\scaleobj{0.97}{\mathbb{Z}}}/3$ | ${\scaleobj{0.97}{\mathbb{Z}}}/3$ | $0$ | $0$ | ${\scaleobj{0.97}{\mathbb{Z}}}/9\oplus{\scaleobj{0.97}{\mathbb{Z}}}/3$

To prove Proposition 6.7, we will need to understand the cohomology of $E^{3}$
as a module over the Steenrod algebra $\mathcal{A}_{3}$, which we recall is
generated by the Steenrod powers $P^{k}$ and the Bockstein operation $\beta$.

######  Lemma 6.8.

The spectrum cohomology of $E^{3}$ is given by

$H^{*}(E^{3};\mathbb{F}_{3})\cong\mathbb{F}_{3}\langle
u\rangle\otimes_{\mathbb{F}_{3}}\mathbb{F}_{3}[\alpha,s]/(\alpha^{2}),\qquad|\alpha|=1,\quad|s|=2,\quad|u|=3(d-p-2),$

with

$P^{k}(u\alpha^{i}s^{j})=\left(\sum_{r=0}^{k}{d-p-2\choose r}{j\choose
k-r}\right)u\alpha^{i}s^{j+2k},\qquad\beta(u\alpha^{i}s^{j})=\left\\{\begin{array}[]{cl}0,&i=0,\\\
-us^{j+1},&i=1.\end{array}\right.$

Moreover $C_{2}=D_{m}/C_{m}$ acts on $H^{*}(E^{3};\mathbb{F}_{3})$ by
$u\alpha^{i}s^{j}\mapsto(-1)^{p+i+j}u\alpha^{i}s^{j}$.

###### Proof.

The key observation to carry out this calculation is that the
$C_{3}$-representation $\psi_{3}\mid_{C_{3}}$ decomposes as $1+\theta$, where
$\theta$ is the $2$-dimensional representation pulled back from the standard
complex $U(1)\cong SO(2)$-representation on
$\mathbb{C}\cong{\scaleobj{0.97}{\mathbb{R}}}^{2}$. In particular, the
associated vector bundle of the representation
$\psi_{m}\otimes(d-p-3+\sigma)\mid_{C_{m}}$ is orientable; write $u\in
H^{*}(E^{3};\mathbb{F}_{3})$ for the corresponding Thom class. The
$\mathbb{F}_{3}$-cohomology of $BC_{3}$ is
$\mathbb{F}_{3}[\alpha,s]/(\alpha^{2})$ with $|\alpha|=1$, $|s|=2$ and

$\beta(\alpha)=s,\qquad P^{k}(\alpha^{i}s^{j})={j\choose
k}\alpha^{i}s^{j+2k}.$

So it remains to understand the action of $\mathcal{A}_{3}$ on $u$. Clearly
$\beta(u)=0$, as if $\beta(u)=nu\alpha$ for some $n\in\mathbb{F}_{3}$, then
$0=\beta^{2}(u)=n^{2}u\alpha^{2}-nus$ and hence $n=0$. For $u_{\theta}$ the
Thom class of $\theta$,
$P^{1}(u_{\theta})=u_{\theta}^{3}=u_{\theta}c_{1}(\theta)^{2}=u_{\theta}s^{2}$.
It then easily follows that $P^{k}(u)={d-p-2\choose k}us^{2k}$.

The residual $D_{m}/C_{m}$-action on the $\mathbb{F}_{3}$-cohomology of
$BC_{3}$ sends $s$ to $-s$, and hence $\alpha$ to $-\alpha$. This action also
switches the orientation of the vector bundle associated to $\theta$, and
hence that of $\psi_{3}=1+\theta$. So that of $\psi_{3}\otimes\sigma$ does not
change then, as $\psi_{3}$ is odd-dimensional. All in all, this means that $u$
is sent by the $C_{2}$-action to $(-1)^{\epsilon}u$ with
$\epsilon=(d+1)+(d-p-3)\equiv p\mod 2$, where the first term is the
contribution of $\mathbb{S}^{(d+1)(\sigma-1)}$. This establishes the claim. ∎

###### Proof of Proposition 6.7.

Consider the $\mathcal{A}_{3}$-submodules of $H^{*}(E^{3};\mathbb{F}_{3})$
given by

$J_{0}:=\langle u\alpha^{i}s^{j}:i+j\equiv 0\mod 2\rangle,\qquad
J_{1}:=\langle u\alpha^{i}s^{j}:i+j\equiv 1\mod 2\rangle.$

Then $H^{*}(E^{3};\mathbb{F}_{3})=J_{0}\oplus J_{1}$ as
$\mathcal{A}_{3}$-modules, and $J_{p}$, where $p$ here is taken mod $2$, is
the $(+1)$-eigenspace of the residual $C_{2}=D_{m}/C_{m}$-action. Therefore
$H^{*}(E^{3}_{hC_{2}};\mathbb{F}_{3})=J_{p}$ as an $\mathcal{A}_{3}$-module,
and the Adams spectral sequence of $J_{p}$ then converges to the stable
homotopy of $E^{3}_{hC_{2}}$. The software [CCBFY22] computes the $E_{2}$-page
of this spectral sequence, from which we determine some of the first homotopy
groups of $E^{3}_{hC_{2}}$. We illustrate the case $d-p=3$ in Figure 5.

Figure 5. $E_{2}$-page of the Adams spectral sequence at the prime $3$ for
$E^{3}_{hC_{2}}$ when $d-p=3$. Here $s$ denotes the degree in the Adams
filtration and $t-s$ is the total degree.

Apart from standard arguments exploting the Leibniz rule and multiplicative
structures of the differentials in Figure 5, we can appeal to some more
refined tricks that allow us to solve the first few possible non-zero
differentials in the case when $p$ is even. Denote by $\widehat{\theta}$ the
$D_{3}$-representation pulled back from the standard $O(2)$-representation on
${\scaleobj{0.97}{\mathbb{R}}}^{2}$. Observe that
$\widehat{\theta}\mid_{C_{3}}\equiv\theta$ in the notation of Lemma 6.8. Write
$\underline{S}(\widehat{\theta}\otimes\sigma\mid_{C_{3}})$ for the unit sphere
bundle of the associated vector bundle
$ED_{3}\times_{C_{3}}(\widehat{\theta}\otimes\sigma)$. As it is an
$S^{1}$-bundle over $BC_{3}$,
$\underline{S}(\widehat{\theta}\otimes\sigma\mid_{C_{3}})$ must itself be a
$K(\pi,1)$, and in fact it must be homotopy equivalent to $S^{1}$ because
$q:\underline{S}(\widehat{\theta}\otimes\sigma\mid_{C_{3}})\to BC_{3}$ does
not admit a section (its euler class is $s\in
H^{2}(BC_{3};\mathbb{F}_{3})=\mathbb{F}[\alpha,s]/(\alpha^{2})$). Moreover the
homology class represented by $q$ is the dual of $\alpha$, and hence the
residual $C_{2}=D_{m}/C_{m}$-action on
$\underline{S}(\widehat{\theta}\otimes\sigma\mid_{C_{3}})\simeq S^{1}$ must
have degree $-1$. So there is an equivalence of unbased spaces
$\underline{S}(\widehat{\theta}\otimes\sigma\mid_{C_{3}})\simeq S^{\sigma}$
which is $C_{2}$-equivariant up to homotopy. We thus get a cofibration

${S^{\sigma}_{+}\simeq\underline{S}(\widehat{\theta}\otimes\sigma\mid_{C_{3}})_{+}}$${(BC_{3})_{+}}$${\mathrm{Th}(\widehat{\theta}\otimes\sigma\mid_{C_{3}})\simeq
S^{-\sigma}\wedge\mathrm{Th}(\psi_{3}\otimes\sigma\mid_{C_{3}})}$$\scriptstyle{q}$

which is $C_{2}$-equivariant up to homotopy (see Notation 5.2($i$)). By
equipping both $S^{\sigma}$ and $BC_{3}$ with distinguished basepoints which
are fixed under the respective involutions and which match under $q$, we can
get rid of the added basepoints and yield a homotopy cofibre sequence of
$C_{2}$-spectra

${\mathbb{S}^{(d+1)(\sigma-1)+2\sigma}}$${S^{(d+1)(\sigma-1)+\sigma}\wedge\Sigma^{\infty}BC_{3}}$${E^{3}.}$$\scriptstyle{q}$

Then, upon inverting $2$ and taking homotopy $C_{2}$-orbits in the sequence
above, we obtain equivalences of spectra

(6.6)
$E^{3}_{hC_{2}}\simeq_{[\frac{1}{2}]}\left\\{\begin{array}[]{cl}(\Sigma^{\infty+1}BC_{3})_{hC_{2}}\simeq\Sigma^{\infty+1}BD_{3},&\text{$d$
even (so $p$ odd),}\\\\[11.38092pt]
\mathrm{hocofib}(q_{hC_{2}}:\mathbb{S}^{1}\to(S^{\sigma}\wedge\Sigma^{\infty}BC_{3})_{hC_{2}}),&\text{$d$
odd (so $p$ even).}\end{array}\right.$

Now by the Kahn–Priddy theorem [KP78] at the prime $3$, the transfer-like map
$\Sigma^{\infty+1}BD_{3}\to\tau_{>1}\mathbb{S}^{1}$ is split surjective on
homotopy groups localised at $3$, and hence by (6.6), the group
$\pi_{*}^{s}(E^{3}_{hC_{2}})_{(3)}$ split surjects onto
$\pi^{s}_{*-1}\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(3)}$ for $*>1$ when $p$
is odd. We will use this fact together with knowledge of $\pi_{*}^{s}$ to
determine the differentials of the red spectral sequence in Figure 5. For
convenience, let us reillustrate a different portion of it in Figure 6.

Figure 6. $E_{2}$-page of Adams spectral sequence at the prime $3$ for
$E^{3}_{hC_{2}}$ when $d-p=3$ and $p$ is odd. Some of the $d_{2}$ (dashed
blue) and $d_{3}$ (dotted green) differentials that will be analysed are
depicted.

The first possible non-zero differential goes from $(12,0)$ to $(11,2)$.
Depending on its (non-)vanishing, we must have that either

$(a)\ \
\pi_{*}^{s}(E^{3}_{hC_{2}})_{(3)}\cong\left\\{\begin{array}[]{cc}{\scaleobj{0.97}{\mathbb{Z}}}/9,&*=11,\\\
{\scaleobj{0.97}{\mathbb{Z}}}/27,&*=12,\end{array}\right.\quad\text{or}\qquad(b)\
\
\pi_{*}^{s}(E^{3}_{hC_{2}})_{(3)}\cong\left\\{\begin{array}[]{cc}{\scaleobj{0.97}{\mathbb{Z}}}/3,&*=11,\\\
{\scaleobj{0.97}{\mathbb{Z}}}/9,&*=12.\end{array}\right.$

Since
$\pi_{10}^{s}\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(3)}\cong{\scaleobj{0.97}{\mathbb{Z}}}/3$,
we must rule out possibility $(a)$ as ${\scaleobj{0.97}{\mathbb{Z}}}/9$ does
not split surject onto ${\scaleobj{0.97}{\mathbb{Z}}}/3$. In other words, the
$d_{2}$ differential in Figure 5 from $(12,0)$ to $(11,2)$ is non-zero and
$(b)$ holds.

The $d_{2}$ differential from $(22,2)$ to $(21,4)$ must be non-zero, and hence
so will that from $(19,1)$ to $(18,3)$ by the multiplicative structure. Indeed
if such differential was zero, it would follow that
$\pi_{21}^{s}(E^{3}_{hC_{2}})_{(3)}\cong{\scaleobj{0.97}{\mathbb{Z}}}/9$,
which does not split surject onto
$\pi_{20}^{s}\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(3)}\cong{\scaleobj{0.97}{\mathbb{Z}}}/3$.

The $d_{2}$ differential from $(24,2)$ to $(23,4)$ must also be non-zero (and
hence that from $(24,3)$ to $(23,5)$) by a similar reason; indeed if it were
trivial, then $\pi_{23}^{s}(E^{3}_{hC_{2}})_{(3)}$ would be isomorphic to
${\scaleobj{0.97}{\mathbb{Z}}}/3\oplus{\scaleobj{0.97}{\mathbb{Z}}}/81$ or
${\scaleobj{0.97}{\mathbb{Z}}}/3\oplus{\scaleobj{0.97}{\mathbb{Z}}}/27$
(depending on whether the $d_{3}$ differential from $(24,2)$ to $(24,5)$
vanishes or not), either of which do not split surject onto
$\pi_{23}^{s}\otimes{\scaleobj{0.97}{\mathbb{Z}}}_{(3)}\cong{\scaleobj{0.97}{\mathbb{Z}}}/3\oplus{\scaleobj{0.97}{\mathbb{Z}}}/9$.

The arguments we just made above, together with standard ones exploiting the
Leibniz rule and the multiplicative structure of the differentials in the
spectral sequence in Figure 5, establish the homotopy groups appearing in
Table 1. ∎

One can keep up using the Kahn–Priddy theorem and go quite far up determining
all possible non-zero differentials when $p$ is odd; we leave it as a fun
exercise to the eager reader. One would also hope that (6.6) could be used in
the case when $p$ is even, but this approach has somehow been inconclusive for
us (at least for the first non-zero differentials that cannot be ruled out by
elementary means).

##  Appendix A Bounded geometry

All throughout, let $M^{d}$ denote a smooth compact manifold (possibly with
boundary) of dimension $d$.

###  A.1. Models for bounded diffeomorphisms and embeddings

Let $N$ be a (possibly non-compact) smooth manifold and fix some smooth
embedding $\iota:M\xhookrightarrow{}N$. For $V\in\mathcal{J}_{0}$, we will
write $\operatorname{Emb}_{\partial}(M\times V,N\times V)$ for the space of
smooth embeddings of $M$ into $N$ that agree with $\iota\times\mathrm{Id}_{V}$
on some neighbourhood of the boundary $\partial M\times V$, endowed with the
Whitney $C^{\infty}$-topology. Following Definition 2.4, the space of bounded
embeddings of $M\times V$ into $N\times V$ relative to $\partial M\times V$ is
the subspace of $[0,+\infty)\times\operatorname{Emb}_{\partial}(M\times
V,N\times V)$ given by

$\operatorname{Emb}_{\partial}^{b}(M\times V,N\times
V):=\left\\{(t,\varphi)\in[0,+\infty)\times\operatorname{Emb}_{\partial}(M\times
V,N\times V):\text{$\varphi$ is $t$-bounded}\right\\}.$

Define similarly its simplicial version
$\operatorname{Emb}_{\partial}^{b}(M\times V,N\times V)$ as in Definition 2.7.
In this section we prove

######  Proposition A.1.

There is a zig-zag of weak equivalences of simplicial group-like monoids

${\mathrm{Diff}^{b}_{\partial}(M\times
V)}$${\cdot}$${\operatorname{Sing}(\mathrm{Diff}_{\partial}^{b}(M\times
V)).}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

Similarly, there is a zig-zag of weak equivalences of simplicial sets

${\operatorname{Emb}^{b}_{\partial}(M\times V,N\times
V)}$${\cdot}$${\operatorname{Sing}(\operatorname{Emb}_{\partial}^{b}(M\times
V,N\times V)).}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

We will only deal with the first part of the statement, as the proof for the
embedding case is completely analogous. Let us first introduce some notation.
Given a topological space $X$, let $\operatorname{Sing}^{\mathsf{col}}(X)$ be
the sub-simplicial set of $\operatorname{Sing}(X)$ consisting of those
singular simplices that satisfy the $\epsilon$-collaring condition of Section
2.2 for some $0<\epsilon<1/2$. Denote by
$\operatorname{Sing}^{\mathsf{col},b}(\mathrm{Diff}_{\partial}(M\times V))$
the sub-simplicial group of
$\operatorname{Sing}^{\mathsf{col}}(\mathrm{Diff}_{\partial}(M\times V))$
consisting of those $j$-simplices which are adjoint to a bounded map
$\Delta^{j}\times M\times V\to M\times V$. Then, there is a zig-zag of maps of
simplicial group-like monoids

(A.1) ${\mathrm{Diff}^{b}_{\partial}(M\times
V)}$${\operatorname{Sing}^{\mathsf{col},b}(\mathrm{Diff}_{\partial}(M\times
V))}$${\operatorname{Sing}^{\mathsf{col}}(\mathrm{Diff}^{b}_{\partial}(M\times
V))}$${\operatorname{Sing}(\mathrm{Diff}^{b}_{\partial}(M\times
V)),}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{3}$

where the map 2 forgets the explicit bounding constant of a simplex. We will
show that all the maps in (A.1) are weak equivalences. We start with 3.

######  Lemma A.2.

The inclusion
$i:\operatorname{Sing}^{\mathsf{col}}(X)\xhookrightarrow{}\operatorname{Sing}(X)$
is a weak equivalence for every topological space $X$.

###### Proof.

We show that the relative homotopy groups
$\pi_{j}(\operatorname{Sing}(X),\operatorname{Sing}^{\mathsf{col}}(X))$ vanish
for all $j\geq 0$. Indeed, a homotopy class
$x\in\pi_{j}(\operatorname{Sing}(X),\operatorname{Sing}^{\mathsf{col}}(X))$
corresponds, by the Yoneda Lemma, to a singular $j$-simplex $g:\Delta^{j}\to
X$ which satisfies the $\epsilon$-collaring condition for all faces
$\sigma\subset\partial\Delta^{j}$ and some $\epsilon>0$. Now fix some
identification
$\Delta^{j}\cong\Delta^{j}\cup_{\partial\Delta^{j}}(\partial\Delta^{j}\times[0,\epsilon])$,
and consider the singular $j$-simplex

$\overline{g}=g\cup(g\mid_{\partial_{\Delta^{j}}}\circ\operatorname{proj}_{\partial\Delta^{j}}):\Delta^{j}\cong\Delta^{j}\cup_{\partial\Delta^{j}}(\partial\Delta^{j}\times[0,\epsilon])\longrightarrow
X.$

By construction $\overline{g}$ now satisfies the $\delta$-collaring conditions
for all faces $\sigma\subset\Delta^{j}$ and some $0<\delta\leq\epsilon$, so
the corresponding relative homotopy class $\overline{x}$ is trivial. But
clearly $g$ and $\overline{g}$ are homotopic relative to the boundary by
shrinking the added collar, and hence $x=\overline{x}=0$ in
$\pi_{j}(\operatorname{Sing}(X),\operatorname{Sing}^{\mathsf{col}}(X))$, as
claimed. ∎

######  Remark A.3.

The inclusion
$i:\operatorname{Sing}^{\mathsf{col}}(X)\xhookrightarrow{}\operatorname{Sing}(X)$
is in fact a simplicial homotopy equivalence; a homotopy inverse is
constructed by induction on the skeleta of $\Delta^{j}$. We will not need this
though.

######  Lemma A.4.

The map $\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}:\operatorname{Sing}^{\mathsf{col}}(\mathrm{Diff}^{b}_{\partial}(M\times
V))\to\operatorname{Sing}^{\mathsf{col},b}(\mathrm{Diff}_{\partial}(M\times
V))$ of (A.1) is a weak equivalence.

###### Proof.

We again show that the relative homotopy groups $\pi_{j}(\leavevmode\hbox
to11.37pt{\vbox to11.37pt{\pgfpicture\makeatletter\hbox{\hskip
5.68657pt\lower-5.68657pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ vanish for all
$j\geq 0$. Such a homotopy $x$ class can be represented, for some
$\epsilon>0$, by an $\epsilon$-collared singular $j$-simplex
$g:\Delta^{j}\to\mathrm{Diff}_{\partial}(M\times V)$, adjoint to a map
$g^{\vee}:\Delta^{j}\times M\times V\to M\times V$ bounded by some $K\geq 0$,
together with a continuous $\epsilon$-collared map
$r:\partial\Delta^{j}\to[0,\infty)$ such that $g(s)$ is $r(s)$-bounded for all
$s\in\partial\Delta^{j}$. To show that $x$ is trivial, we need to extend $r$
to a continuous $\delta$-collared map $R:\Delta^{j}\to[0,\infty)$, for some
$0<\delta\leq\epsilon$, such that $g(s)$ is $R(s)$-bounded for all
$s\in\Delta^{j}$. Fix some identification
$\Delta^{j}\cong(\partial\Delta^{j}\times[0,\epsilon])\cup_{\partial\Delta^{j}\times\\{\epsilon\\}}\Delta^{j}$;
then $R\mid_{\partial\Delta^{j}\times[0,\epsilon/2]}\equiv
r\circ\operatorname{proj}_{\partial\Delta^{j}}$ whilst
$R\mid_{\partial\Delta^{j}\times[\epsilon/2,\epsilon]}$ is a linear
interpolation along $[\epsilon/2,\epsilon]$ between $r$ and the constant map
$c_{K}:\partial\Delta^{j}\times\\{\epsilon\\}\to[0,\infty)$ with value $K\geq
0$. Finally set $R$ to be constant of value $K$ in the inner
$\Delta^{j}\subset(\partial\Delta^{j}\times[0,\epsilon])\cup_{\partial\Delta^{j}\times\\{\epsilon\\}}\Delta^{j}$.
Then $R$ is as required, and hence the relative homotopy class
$x\in\pi_{j}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ is trivial. ∎

######  Remark A.5.

For $CAT=\mathrm{Top}$, the map 1 of (A.1) is an equality and thus, at this
point, Proposition A.1 is established in the topological case.

###### Proof of Proposition A.1.

It remains to show that 1 is a weak equivalence, i.e., that the relative
homotopy groups $\pi_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ vanish for all
$k\geq 0$. This is clear for $k=0$ by definition. Such a homotopy class in
$\pi_{k}(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ is represented by a
bounded homeomorphism

$\overline{g}=(\mathrm{proj}_{\Delta^{k}},g):\Delta^{k}\times M\times
V\longrightarrow\Delta^{k}\times M\times V$

which is collared in the simplex direction and such that
$g\mid_{\partial\Delta^{k}\times M\times V}$ is smooth. Therefore $g$ is
smooth on (a neighbourhood of) $\partial\Delta^{k}\times M\times V$. We need
to smooth $g$ outside of such neighbourhood in the $\Delta^{k}$-direction and
preserving boundedness. For $r\in\Delta^{k}$, we will write
$g_{r}\in\mathrm{Diff}_{\partial}(M\times V)$ for $g\mid_{\\{r\\}\times
M\times V}$.

Standard smoothing techniques [Mun66, §4] (see also [Kup19, Prop. 6.4.2] or
[Lur09, Prop. 1]) can be used to prove the following: given nested compact
subsets $L\subset K\subset\Delta^{k}\times M\times V$ with
$L\subset\mathrm{int}\ K$ and any arbitrarily small $\epsilon>0$, there exists
a homotopy $H:I\times\Delta^{k}\times M\times V\to M\times V$ from $g$ to some
map $g^{\prime}:\Delta^{k}\times M\times V\to M\times V$ satisfying that:

  * $(i)$

$H$ remains fixed on $\Delta^{k}\times M\times V-\mathrm{int}\ L$. In
particular $g$ and $g^{\prime}$ agree there.

  * $(ii)$