5_2_slice.tex

In this section, we discuss the slice spectral sequence, the equivariant version of the Postnikov spectral
sequence. Very little is known about the slice spectral sequence, so you could pursue it in your research. For
example, its use in~\cite{HHR} depended on an explicit identification of the slices with something else, which does
not generalize.

\subsection*{Postnikov towers.} For this section, we work in the category $\Top$ of spaces. We want a collection of
functors $P_n\colon\Top\to\Top$, called \term[Postnikov section]{Postnikov sections}, together with natural
transformations $P_n\to P_{n-1}$, such that
\begin{itemize}
	\item
	\[\pi_i(P_nX) = \begin{cases}
		\pi_i(X), &0\le i\le n\\
		0, &\text{otherwise;}
	\end{cases}\]
	\item $X\cong\holim P_nX$; and
	\item the homotopy fiber of $P_nX\to P_{n-1}X$ is weakly equivalent to $K(\pi_n(X), n)$.
\end{itemize}
So we obtain a \term{Postnikov tower} $\dotsb\to P_2X\to P_1X\to P_0X$, which is in a sense dual to the cellular
filtration.\index{cellular filtration}
\begin{rem}
The axiomatization of the idea of a Postnikov tower in a triangulated category is called a
\term[t-structure@$t$-structure]{$t$-structure}~\cite{BBD}. The dual notion of a cellular filtration is axiomatized
as a \term{weight structure}~\cite{Bondarko}.
\end{rem}
These $P_nX$ have only finitely many nonzero homotopy groups, hence must be very large spaces. There are multiple
different point-set models for them. Here's one model.

Choose an $\alpha\in\pi_{n+1}(X)$, which defines a homotopy class of maps $f_\alpha\colon S^{n+1}\to X$. Let
$\widetilde X$ be the pushout
\[\xymatrix{
	S^{n+1}\ar[r]^{f_\alpha}\ar[d] & X\ar[d]\\
	D^{n+2}\ar[r] & \widetilde X.\pushout
}\]
This kills $\alpha$, and since $S^{n+1}$ is $n$-connected, the induced map $\pi_k(X)\to\pi_k(\widetilde X)$ is an
isomorphism for $k < n+1$. Then we can iterate.

This has a fatal flaw: it's not functorial. As usual, we fix this with the small object argument. Consider
\emph{all} of the maps $S^{n+1}\to X$ and take the pushout\index{small object argument}
\[\xymatrix{
	\bigvee S^{n+1}\ar[r]\ar[d] & X\ar[d]\\
	\bigvee D^{n+2}\ar[r] & X_{n+1}.
}\]
Continue this way for maps $S^N\to X$ for $N\ge n+1$ and let $P_nX\coloneqq \colim_N X_N$. This is functorial.

The functor $P_n$ can be described as localization, e.g.\ in~\cite{MandellShipley}, which allows for some slick
high-tech setups: if you work with presentable $\infty$-categories, you can describe $P_n$ as adjoint to the
inclusion of spaces with homotopy groups within $[0,n]$, as in~\cite[\S5.5.6]{HTT}.
\index{presentable $\infty$-category}

The Postnikov tower leads to the \term{Atiyah-Hirzebruch spectral sequence} (AHSS), which for a generalized
homology theory $E$, has signature
\[E_{p,q}^2 = H_p(X; E_q(*))\Longrightarrow E_{p+q}(X).\]
\begin{ex}[Maunder~\cite{Maunder}]
If you play the same game with the CW filtration, you obtain an isomorphic spectral sequence. It's a good exercise
to work this out yourself.
\end{ex}
This works in more than just spaces: you can set it up for spectra, and (with a little technical work) for
(commutative) ring spectra.
\subsection*{The equivariant case.}
We'd like to do this in $\Spc^G$. The slice spectral sequence will be an analogue to the Atiyah-Hirzebruch spectral
sequence. One difficulty will be understanding the associated graded, which is considerably more complicated than
the nonequivariant Postnikov or CW associated graded complexes, and this is ultimately because there are more
spheres around.

The first piece of the slice spectral sequence was worked out by Dugger~\cite{DuggerKR}. His motivation was the
analogy between $C_2$-equivariant homotopy theory and motivic homotopy theory. Following
Bloch-Lichtenbaum~\cite{BlochLichtenbaum} and Grayson~\cite{Grayson}, the goal was to approach the
Quillen-Lichtenbaum conjecture~\cite{Quillen}, the existence of a spectral sequence\index{motivic homotopy
theory!analogy with $C_2$-equivariant homotopy theory}\index{Quillen-Lichtenbaum conjecture}
\[H^p(X; \Z(-\e/2))\Longrightarrow K^{p+q}(X).\]
Here, $H^*(\bl,\Z(-\e/2))$ is ``motivic cohomology,'' which was not well-understood when the conjecture was
formulated. The $-\e/2$ is a \term{Tate twist}, which is akin to Bott periodicity for algebraic $K$-theory of
finite fields. See~\cite{BlochLichtenbaum, Grayson} for details or Mitchell~\cite{Mitchell} for an exposition.
Dugger replaced this with a conditionally convergent spectral sequence
\[E_2^{p,q} = H^{p,r-q/2}(X;\underline\Z)\Longrightarrow \KR^{p+q,r}(X),\]
where
\begin{itemize}
	\item $X$ is any $C_2$-space,
	\item $\underline\Z$ is the constant Mackey functor valued in $\Z$, and
	\item $\KR$ is Atiyah's $\KR$-theory~\cite{AtiyahKR}.
\end{itemize}
If $q$ is odd, we take $E_2^{p,q} = 0$.
\index{Bott periodicity}\index{motivic cohomology}\index{K-theory@$K$-theory!real equivariant $K$-theory}

The general formulation of the slice spectral sequence was worked out in~\cite{HHR}; see also
Hu-Kriz-Ormsby~\cite{HKO11}. The exposition in~\cite{HHR} is pretty good, and you should also check out Mike Hill's
introduction~\cite{HillSlice}.

To get at the spectral sequence, we first approach the filtration. The motivation is to have a Postnikov section
$P_H$ for the regular representation $\rho_H$ of $H$. The \term{slice cells} will be the cells\index{regular
representation}
\[\set{G_+\wedge_H S^{m\rho_H}, \Sigma^{-1}G_+\wedge_H S^{m\rho_H}}.\]
The \term[dimension!of a slice cell]{dimensions} of these slice cells are $m\abs H$, resp.\ $m\abs H - 1$.

Slice cells are well-behaved under the ``change functors'' $i_K^*$, $G_+\wedge_K\bl$, and $N_K^G$: all of these
preserve slice cells.
\begin{defn}
Let $X\in\Spc^G$.
\begin{itemize}
	\item $X$ is \term{slice $n$-null} if $\Map(\widehat S, X)$ is contractible (as a $G$-space) for all slice
	cells $\widehat S$ of dimension greater than $n$. One also says $X$ is \term{slice $< n$} or
	\term{slice $\le n-1$}.
	\item $X$ is \term{slice $n$-positive} (also \term{slice $> n$} or \term*{slice $\ge n+1$}) if $\Map(\widehat S,
	X)$ is contractible (as a $G$-space) for all slice cells $\widehat S$ of dimension at most $n$.
\end{itemize}
\end{defn}
\begin{ex}[\cite{HHR}, Prop.~4.11]
Let $X$ be a $G$-spectrum.
\label{slicerecognition}
\begin{enumerate}
	\item Show that $X$ is slice $0$-positive iff it's $-1$-connected.\footnote{By \term*{$n$-connected} we mean
	the Mackey functor $\underline\pi_kX = 0$ for $k\le n$, and by \term*{$n$-coconnected} we mean
	$\underline\pi_kX = 0$ for $k\ge n$.}
	\item Show that $X$ is slice $-1$-positive iff it's $-2$-connected.
	\item Show that $X$ is slice $0$-null iff it's $0$-coconnected.
	\item Show that $X$ is slice $-1$-null iff it's $-1$-coconnected.
\end{enumerate}
\end{ex}
\index{n-connected@$n$-connected}
\index{n-coconnected@$n$-coconnected}
\index{coconnected|see {$n$-coconnected}}
\begin{prop}[\cite{HHR}, Prop.~4.15]
$X$ is slice $n$-positive iff up to weak equivalence, there's a filtration $X_0\subseteq
X_1\subseteq\dotsb\subseteq X$ such that $X_i/X_{i-1}$ is a wedge of slice cells of dimensions greater than $n$.
\end{prop}

We localize at the slices of dimension greater than $n$ to obtain $P^nX$, the
\term[n-slice section@$n$-slice section]{$n$-slice section} of $X$. This is a bit tricky, because there are maps
$S^V\to S^W$ when $V\subseteq W$ that aren't null-homotopic, frustrating our approach to defining the Postnikov
tower, but the key is that if $V$ contains a trivial representation, all such maps are
null-homotopic.\index{trivial representation}

Localization means there's a map $X\to P^nX$ by fiat, and one can show that the homotopy fiber of this map is slice
$n$-positive. We obtain a tower $\dotsb\to P^nX\to P^{n-1}X\to \dotsb$. Let $P_n^nX$ denote the homotopy fiber of
$P^nX\to P^{n-1}X$; $P_n^nX$ is called the \term[n-slice@$n$-slice]{$n$-slice} of $X$.
\begin{ex}[\cite{HHR}, Prop.~4.20]
Show that the $-1$-slice of $X$ is
\[P_{-1}^{-1}X = \Sigma^{-1} H(\underline\pi_{-1}X),\]
where again $\underline\pi_{-1}$ means to take the homotopy group as a Mackey functor. (Hint: use
\cref{slicerecognition}.)
\end{ex}
\cite{HHR} heavily use the fact that the regular\index{regular representation}
representation contains a copy of all irreducibles to understand what's going on, which is great for finite groups,
but doesn't work for compact Lie groups. There's a different perspective adopted by Dugger, that $G/H_+\wedge S^n$
detects homotopies, so we can restrict to slice cells $G/H_+\wedge S^V$ where $V$ contains a copy of the trivial
representation (and $\Sigma^{-1}$ of these cells). The slice filtration has some other nice properties: $\holim
P^nX\cong X$, and $\colim P^nX\simeq *$.

These localizations are controlled by the subcategory of $\Spc^G$ determined by positive shifts and cofibers of
slice cells. Negative shifts are not allowed, so when passing to the homotopy category, this isn't a triangulated
subcategory.

This slice filtration produces the \term{slice spectral sequence}, which is a strongly convergent spectral sequence
\begin{equation}
\label{SlSS}
E_2^{s,t} = \pi_s^G P_t^tX\Longrightarrow \pi_{s-t}^G X.
\end{equation}
Here we use the Adams grading, so this may look funny compared to the usual grading on, e.g.\ the Serre spectral
sequence. You can refine the slice spectral sequence into a spectral sequence of Mackey functors, and there is an
$\RO(G)$-graded version. If you think about it, you'll see this is a first- and third-quadrant spectral
sequence.\index{Adams grading}\index{Mackey functor}

The key to understanding the slice spectral sequence is understanding the slices $P_t^tX$, and this is hard:
\cite{HHR} uses the fact that real bordism has nice slices, which does not generalize. Nonetheless, there are some
general results known: Hill-Yarnall~\cite{HillYarnall} provide a computable condition for slice $n$-connectivity,
and Wilson~\cite{Wil17} provides an algebraic description of the category of $n$-slices for $G$ finite.
% TODO: study of slice connectivitydue to HillYarnall
% TODO: algebraic characterization due to Wilson

There are some computations in the literature: Dugger~\cite{DuggerKR}, Hill-Meier~\cite{HillMeier}, and
Greenlees~\cite{GreenCalc} work out some over $C_2$; Hill~\cite{HillRealBordism} and
Hill-Hopkins-Ravenel~\cite{HHR_C4} study the slices of $N_{C_2}^{C_{2^k}}\MR$; and Yarnall~\cite{Yarnall} and
Hill-Hopkins-Ravenel~\cite{HHR_HZ} study the slices of $\Sigma^V H\underline\Z$.

\index{real bordism}

%
%This concludes the content of the course; on Thursday there will be donuts and discussion of future research. Some
%of these are important open questions that may be intractable, and others could be research projects; hopefully
%it will be clear which is which.

These notes will hopefully be a living document; feel free to update with better explanations, or to fix mistakes,
or to add examples, or anything like that. There are relatively few references for equivariant homotopy theory, and
it would be great for the notes to be one more, hopefully in a presentable way.