2_21_wirthmuller.tex

\begin{quote}
	\textit{``What do homotopy theorists do when changing planes?''\\
	``Look for a transfer map.''}
\end{quote}
In this section, we'll conver the Wirthmüller isomorphism, the equivariant analogue of the nonequivariant statement
that in the stable homotopy category $\Ho(\Spc)$, finite sums and products are equivalent:
\begin{equation}
\label{additivespec}
\bigvee_{i=1}^n X_i\stackrel\cong\longrightarrow \prod_{i=1}^n X_i.
\end{equation}
This is a backbone of the stable category: among other things, it ensures $\Ho(\Spc)$ is additive. We'll assume $G$
is a finite group; there's a statement for compact Lie groups, but it's more complicated.

Tom Dieck splitting is another important structural result about $\pi_G^*(\Sigma^\infty X)$. When $X = S^0$, this
is used to compute the equivariant stable homotopy groups of the spheres (which is, of course, not completely
known). This can be used to recover the Burnside category.\index{Burnside category}

Transfers play a key role in both of these results, somewhat implicitly in the Wirthmüller isomorphism, but very
explicitly in tom Dieck splitting.
\begin{thm}[Wirthmüller isomorphism]
\label{Wirthiso}
\index{Wirthmüller isomorphism}\index{pistar-isomorphism@$\pi_*$-isomorphism}
Let $H$ be a subgroup of $G$ and $X$ be an object of $\Sp^H$.\footnote{Here, we're implicitly using the complete
universe $U$ to define $\Spc^G$ and $\Spc^H$.\index{complete universe}} Then, there is a $\pi_*$-isomorphism
$G\wedge_H X\simeqto F_H(G,X)$.
\end{thm}
\begin{ex}
Adapt the proof of~\eqref{additivespec} for nonequivariant spectra to show that the natural map from finite sums to
finite products in $\Spc^G$ is also an isomorphism.
\end{ex}
However, if you only have~\eqref{additivespec}, you've described the category of $G$-spectra structured by a
universe with only trivial representations!

Applying \cref{Wirthiso} to $X = S^0$ computes the Spanier-Whitehead duals of orbits.
\begin{cor}
\label{wirthcor}
In the situation of \cref{Wirthiso}, $\Sigma^\infty(G/H)_+\cong F_H(G,S^0)\cong F((G/H)_+, S^0)$.
\end{cor}
That is, $\Sigma^\infty (G/H)_+$ is its own Spanier-Whitehead dual. The slogan is that orbits are self-dual in the
equivariant stable category when $G$ is finite.\footnote{When $G$ is a compact Lie group, there's a degree shift
arising from the tangent representation of $G$ on the Lie algebra of $H$. \TODO: double-check
this.}\index{Spanier-Whitehead duality}

Recall that the forgetful map $i_H^*\colon \Spc^G\to\Spc^H$ has a left and a right adjoint, respectively
$G\wedge_H\bl$ and $F_H(G,\bl)$. These spectrum-level constructions are induced by applying the space-level
functors levelwise.
\begin{ex}
Show that these definitions of $G\wedge_H\bl$ and $F_H(G,\bl)$ are consistent with the structure maps.
\end{ex}
There are three approaches to proving \cref{Wirthiso}.
\begin{enumerate}
	\item One is a connectivity argument: show that on the space level, the maps get more and more highly
	connected.
	\item\label{transferproof} Another approach is to construct an explicit inverse, using a transfer map.
	\item One can also set up a Grothendieck six-functor formalism to prove it, which sets up a general theory for
	when a lax monoidal functor's left and right adjoints coincide. See~\cite{FHM, WirthRevisited, BDS} for a proof
	using this approach; the first and the third papers set up general theory, and the second shows that it applies
	to the Wirthmüller map. This makes interesting contact with base-change theory in algebraic
	geometry.\index{six-functor formalism}
\end{enumerate}
\begin{warn}
The proof of \cref{Wirthiso} given in~\cite{LMS} is incorrect, and the fix is nontrivial. It takes
approach~\eqref{transferproof}.
\end{warn}
We'll use a connectivity argument, which will introduce some tools we'll find useful later.
\begin{proof}[Proof of \cref{Wirthiso} ($X$ connected)]
Though we assume $X$ is connected, the same proof can be adapted when $X$ is bounded-below. The theorem is true for
general $X$, but this proof may not work in that case.

Let $X\in H\Top_*$. Then, there's a map $\theta\colon G\wedge_H X\to F_H(G,X)$ defined by
\[\theta(g_1,x)(g_2) = \begin{cases}
	g_2g_1x, &g_2g_1\in H\\
	*, &\text{otherwise.}
\end{cases}\]
\begin{ex}
Show that $\theta$ is a $G$-map.
\end{ex}
$\theta$ induces a map $\overline\theta\colon G\wedge_H X\to F_H(G,X)$ in $\Spc^G$. We'll show that it's an
equivalence by computing the connectivity of $\theta$.

Let $K\subseteq G$, $\rho$ be the regular representation of $G$, and $m\in\N$. Then, we'll compute the connectivity
of\index{regular representation}
\[\theta_{m,K}\colon \paren{G\wedge_H \Sigma^{m\rho}X}^K\longrightarrow \paren{G_H(G, \Sigma^{m\rho} X)}^K.\]
When $G$ is finite, the sequence $(\rho, 2\rho, 3\rho,\dotsc)$ is cofinal (i.e.\ the colimits are the same) in the
filtered diagram of finite-dimensional representations of the complete universe $U$. Thus, to understand
$\colim_{V\subset U} \Omega^V\Sigma^V X$, it suffices to understand what happens when $V = m\rho$.
\begin{ex}
The reason this is true is that when $G$ is finite, the regular representation $\rho$ contains a copy of every
irreducible as a summand. Prove this by doing a character computation.\index{regular representation}
\end{ex}
We'll show the connectivity of $\theta_{m,K}$ is increasing in $m$, which means that
$\pi^K_*\overline\theta\colon\pi_*^K(G\wedge_H X)\to\pi_*^K(F_H(G,X))$ is an isomorphism.

The calculation itself will use the $K, H$ double coset decomposition of $G$ to identify the $K$-fixed points as a
sum and as a product, and we understand the connectivity of both of these from nonequivariant homotopy
theory.\index{double cosets}

Let $S\subset G$ be a set of
representatives for the double coset partition of $G$, i.e.\ under the $K\times H$-action $(k,h)\cdot g =
k^{-1}gh$. Then,
\begin{equation}
\label{GZKfixed}
\paren{G\wedge_H Z}^K = \paren{\coprod_{g\in S} (KgH)_+\wedge_H Z}^K.
\end{equation}
Either $g^{-1}Kg\subseteq H$ or it isn't.
\begin{itemize}
	\item If $g^{-1}Kg\subseteq H$, then the action is only in $Z$, so $((KgH)_+\wedge_H Z)^K = Z^{g^{-1}Kg}$.
	\item If $g^{-1}Kg\not\subseteq H$, then the action is free, so $((KgH)_+\wedge_H Z)^K = *$.
\end{itemize}
In particular~\eqref{GZKfixed} simplifies to
\[\paren{G\wedge_H Z}^K\cong \bigvee_{\substack{g\in S\\g^{-1}Kg\subseteq H}} Z^{g^{-1}Kg}.\]
Next we look at $(F_H(G,Z))^K$. If $S$ is a set of representatives of the double cosets, so is
$S^{-1}\coloneqq \set{g^{-1}\mid g\in S}$, so we can decompose
\begin{align*}
\paren{F_H(G,Z)}^K &\cong \paren{F_H\paren{\coprod_{g\in S} Kg^{-1}H, Z}}^K\\
&\cong \prod_{g\in S} F_H(Kg^{-1}H, Z)^K\\
&\cong \prod_{g\in S} Z^{(gKg^{-1})\cap H} \cong \prod_{g\in S} Z^{(g^{-1}Kg)\cap H}.
\end{align*}
This identification sends $f\mapsto\set{f(g)}$; the last equivalence is by using $S^{-1}$ as the set of
representatives instead of $S$.

In particular, the space-level Wirthmüller isomorphism may be written
\begin{equation}
\label{fixedwirth}
\bigvee_{\substack{g\in S\\g^{-1}Kg\subseteq H}} Z^{g^{-1}Kg}\stackrel\cong\longrightarrow \prod_{g\in S}
Z^{(g^{-1}Kg)\cap H}.
\end{equation}
This is nice-looking, but the indexing sets are slightly different. To overcome this, we'll
factor~\eqref{fixedwirth} as
\[\xymatrix{
	\bigvee_{\substack{g\in S\\g^{-1}Kg\subseteq H}} Z^{g^{-1}Kg}	\ar[r]^-{\vp_1}
	& \prod_{\substack{g\in S\\g^{-1}Kg\subseteq H}} Z^{(g^{-1}Kg)\cap H}\ar[r]^-{\vp_2}
	& \prod_{g\in S} Z^{(g^{-1}Kg)\cap H},
}\]
where $\vp_1$ is the natural map from the sum to the product and $\vp_2$ is inclusion. Then, we will estimate the
connectivity of $\vp_1$ and $\vp_2$ separately in the case when $Z = \Sigma^{m\rho}X$. Namely, we'd like to show
that they're both $(m[G:K] + O(1))$-connected.

First, what are the fixed points in $(S^{m\rho})^{g^{-1}Kg}$? This is a sphere whose dimension has been shrunk by
$K$, so smashing with it produces something $m[G:K]$-connected. Then, the usual argument about turning sums into
products says that $\vp_1$ is about $2m[G:K]$-connected.

For $\vp_2$, what's the connectivity of the missing factors in the domain? In this case, the connectivity is about
$m[G:K]$.

Thus, the connectivity of $\theta_{m,K}$ is $m([G,K] + O(1))$,\footnote{\TODO: I want to run through the
connectivity carefully and ensure I made no typos.} so the spectrum-level map $\overline\theta$ is a
$\pi_*^K$-isomorphism. More precisely, the cofiber of $\theta\colon G\wedge_H X\to F_H(G,X)$ has trivial $\pi_*^K$,
and there's a little bit to do here to check this.
\end{proof}
\begin{rem}
Let's see how this connects to the transfer. Let $V$ be a $G$-representation such that there's an embedding
$G/H\inj V$. Then, we obtain an $H$-map $G\wedge_H S^V\to S^V$ that takes points in $G\setminus H$ to the
basepoint. The adjoint of this map is the $G$-map $G\wedge_H S^V\to F_H(G,S^V)$.

On the other hand, we have a Pontrjagin-Thom map $S^V\to G\wedge_H S^V$ induced as follows: $G/H\inj V$ induces a
map $G\wedge_H D(V)\to V$, and therefore a map $S^V\to G\wedge_H D(V)/(G\wedge_H S(V))\cong G\wedge_H S^V$. The
observation is that the composition $S^V\to G\wedge_H S^V\to S^V$ is the identity (you can and should think about
this: it's possible to write down an explicit homotopy inverse). This motivates the slogan that ``the Wirthmüller
isomorphism is the inverse of the transfer map,'' which we'll say more about later.\index{Pontrjagin-Thom
construction}\index{Wirthmüller isomorphism!as an inverse of the transfer map}
\end{rem}
You can use \cref{wirthcor} to explicitly write down the transfer map: given subgroups $K\subseteq H\subseteq G$,
we have the ``right-way'' map $G/H\to G/K$, and therefore a map $\sus(G/K_+)\cong D(G/K_+)\to
D(G/H_+)\cong\sus(G/H_+)$. By the Wirthmüller isomorphism, this is the same as a map $F(G/K_+,S)\to F(G/H_+,S)$,
which is exactly the transfer map!\index{transfer map!from the Wirthmüller isomorphism}

This is a little circular; if you look carefully into the proof of \cref{Wirthiso}, the transfer map is already
there. But the point is more philosophical: the existence of transfers is the same thing as the Wirthmüller
isomorphism.
\begin{rem}
Transfer maps can be studied in more generality, e.g.\ in the context of equivariant vector bundles on homogeneous
spaces. Rothenberg wrote some stuff, but~\cite{LMS} is probably the best source.
\end{rem}