Relationship between Smith's special homology groups and equivariant homology theory

Question

EDIT: Tyler Lawson's answer was so nice that I was inspired to rewrite the notes discussed below to use Bredon homology in the definition of the Smith special homology groups. The original version is available for readers who prefer being low-tech; just click on the abstract button and a download button for this will become available.

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Let $p$ be a prime, let $G$ be a finite $p$-group, and let $X$ be a "reasonable" finite-dimensional $G$-space (let me be vague about what reasonable means, but certainly a finite-dimensional simplicial complex upon which $G$ acts simplicially counts). Around 1940, P. Smith proved a number of theorems that relate the algebraic topology of $X$ to the algebraic topology of the fixed point set $X^G$. For instance, one of his theorems says that if $X$ is mod-$p$ acyclic, then so is $X^G$ (which implies in particular that $X^G$ is nonempty). I have notes on Smith theory on my page of notes here that give more details.

The original proofs of the main results of Smith theory use the "special homology groups" of Smith, which are defined algebraically and whose geometric meaning seems to me to be quite mysterious. I describe these homology groups in the notes above. When Borel introduced equivariant homology via the Borel construction, one of his original applications was to give a new proof of Smith's main results. These days, equivariant homology (both in the original Borel flavor and also using Bredon's more powerful definitions) is one of the fundamental tools in the theory of transformation groups.

Question: Is there a way to relate Smith's special homology groups to these modern developments? How are they related to equivariant homology? From my reading of the literature, they seem to have been mostly discarded from the toolbox of people working in the subject. Is it now understood that whatever information they contain is also contained in other places?

Answer 1

Smith's special homology groups are special instances of Bredon homology. Here's roughly how it goes, taken from Peter May's "A generalization of Smith theory".

A Bredon coefficient system $M$ is some kind of functorial assignment of abelian groups to $G$-sets of the form $G/H$, or some kind of functor from $G$-sets to abelian groups that takes disjoint unions to direct sums. In particular, when $G = \Bbb Z/p$ we unravel this and find that it an abelian group $A$ with $G$-action, a second abelian group $B$ acted on trivially, and a $G$-map $A \to B$. Associated to a simplicial complex with "good" $G$-action (the stabilizer of a simplex fixes the simplex--this is equivalent to your description in terms of $X/G$), you get a complex computing the Bredon homology groups $H_*(X;M)$.

Here are some Bredon coefficient systems:

• The trivial map $0 \to \Bbb F_p$. This computes the homology $H_*(X^G;\Bbb F_p)$.

• The identity map $\Bbb F_p \to \Bbb F_p$. This computes the homology $H_*(X/G;\Bbb F_p)$.

• The augmentation $\Bbb F_p[G] \to \Bbb F_p$. This computes the homology $H_*(X;\Bbb F_p)$.

• Let $\tau = 1 - t$ where $t$ is the generator of $\Bbb Z/p$. Then there's a sub-coefficient-system $\tau^k \Bbb F_p[G] \to 0$ of the previous one (it's the $k$'th power of the augmentation ideal, because $\tau$ generates the augmentation ideal). These Bredon homology groups compute Smith's special homology groups.

Most of the manipulations that one does to obtain the results of Smith theory are specializations of standard relationships and exact sequences that one gets from functoriality and long exact sequences for Bredon homology. It simply expresses the proof that you already wrote in slightly more standardized language.

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You also asked about relationships to modern developments. Bredon (co)homology is certainly part of the standard toolbox in equivariant homotopy theory. In principle, these homology and cohomology theories are naturally part of equivariant stable homotopy theory. One reason why you don't see Bredon theory mentioned as often anymore is that much of the research focus in equivariant stable theory is on studying the "genuine" equivariant stable category rather than the "naive" stable category (don't blame me for the naming), because things like transfer maps and duality theorems live in the former rather than the latter.

Bredon cohomology theories are residents of the naive stable category. They only come from the genuine stable category if they have transfer maps making them into something called Mackey functors. I get a little bit confused about the difference between Bredon homology and cohomology, but unless I've made a mistake the cohomology version of the above argument does lift to Mackey functors and so it should be something that can be phrased in terms of the genuine stable category. (I'm being cagey because I've been burned about subtle issues on similar questions before and I haven't verified things carefully.)

I don't have a better explanation why there aren't as many new developments in Smith theory--in particular, whether it's a problem of interest or generalizability.