Equivariant cohomology of fixed points using the localisation theorem

Question

I am trying to understand the Smith-Thom inequality for spaces equipped with an action by a cyclic group and also the case, when it's an equality:

In the following, let $G=\mathbb{Z}/p$, $\mathbb{F}$ be a prime field of characteristic $p$ and $H_G^*(-)$ be the Borel construction for equivariant cohomology with coefficients in $\mathbb{F}$, i.e. $H_G^i(X):=H^i((X\times EG)/G;\mathbb{F})$ for a $G$-space $X$. I want to understand why the following theorem holds:

Theorem 1 (Smith-Thom inequality) Let $X$ be a finite $n$-dimensional $G$-complex of finite orbit type. Then $\sum_i\dim_{\mathbb{F}}H^i(X^G)\leq\sum_i \dim_{\mathbb{F}}H^i(X)$, with equality if and only if the Serre spectral sequence of $X_G\to BG$ collapses at $E_2$ and $G$ acts trivially on $H^*(X)$.

In the literature, I have seen this stated as a direct consequence of the localisation theorem:

Localisation theorem Let $S=\{t^k\}_k$ where $t\in H^2(BS^1)$ is a generator. If $X$ is a finite dimensional $G$-complex, the inclusion $X^G\to X$ induces an isomorphism $$S^{-1}H^\ast_G(X)\to S^{-1}H^\ast_G(X^G).$$

For example in Hsiangs' book Cohomology theory of topological transformation groups, Theorem 1 is stated as a corollary of the Localisation Theorem (Corollary 2, Chapter IV.1, page 46)

There is also a proof given in Prop. III.4.16 in Tom Diecks Transformation groups using the following statement (Prop. III.4.9 in Tom Diecks' book):

Theorem 2 Let $X$ be a finite $n$-dimensional $G$-complex of finite orbit type. Then $H_G^k(X^G)\cong H^k_G(X)$ for all $k>n$.

In chapter III.4, Tom Dieck derives Theorem 2 from the localisation theorem by using a Gysin Sequence associated to the standard vector bundle (as he calls it) $$EG\times_G \mathbb{C}\to BG$$ which gives rise to an exact sequence $$H_G^{r+1}(X\times S^1)\to H^r_G(X)\xrightarrow{\cup} H^{r+2}_G(X)\to H^{r+2}_G(X\times S^1)$$

I have the following question(s).

Question

1. Where does the last exact sequence come from?
2. Is there a different proof of Theorem 1, which does not use this standard vector bundle?

Answer 1

I am just posting my comments as one answer.

Let $\mathbb{S}^1$ denote the one-dimensional circle Lie group. The classifying space $B\mathbb{S}^1$ is simply connected with integral cohomology ring isomorphic to the polynomial ring $\mathbb{Z}[c_1]$, where $c_1$ is the element of $H^2(B\mathbb{S}^1;\mathbb{Z})$ representing the first Chern class of the associated $\mathbb{C}$-line bundle over $B\mathbb{S}^1$ induced from the universal principal $\mathbb{S}^1$-bundle, $$\pi:E\mathbb{S}^1\to B\mathbb{S}^1.$$
In particular, $R\pi_*\underline{\mathbb{Z}}$ is quasi-isomorphic to a two-term complex of Abelian sheaves concentrated in degrees $0$ and $1$ whose zeroth cohomology sheaf and first cohomology sheaf are both isomorphic to the locally constant sheaf $\underline{\mathbb{Z}}$ on $B\mathbb{S}^1$ (the zeroth cohomology sheaf is canonically isomorphic to this, but the isomorphism of the first cohomology depends on an orientation of $\mathbb{S}^1$). The induced Leray–Serre Spectral Sequence for $\pi$ reduces to a long exact sequence, usually called the Thom–Gysin Sequence of this principal $\mathbb{S}^1$-bundle, for all $n\geq 1$. $$\dotsb \to H^{n-1}(B\mathbb{S}^1;\mathbb{Z}) \xrightarrow{c_1 \cup -} H^{n+1}(B\mathbb{S}^1;\mathbb{Z}) \to H^{n+1}(E\mathbb{S}^1;\mathbb{Z}) \to \dotsb. $$ Of course the total space $E\mathbb{S}^1$ is connected and contractible, so the long exact sequence reduces to the evident isomorphisms for $n=2m+1$ odd. $$\mathbb{Z}\cdot c_1^{m} \xrightarrow{c_1\cup -} \mathbb{Z}\cdot c_1^{m+1}.$$

Anyway, for every CW complex $B$, for every principal $\mathbb{S}^1$-bundle over $B$, $$\rho:E \to B,$$ there exists a continuous function $f_\rho:B\to B\mathbb{S}^1$ (unique up to homotopy) such that the pullback via $f_\rho$ of the universal principal $\mathbb{S}^1$-bundle $E\mathbb{S}^1$ is isomorphic to $E$ as a principal $\mathbb{S}^1$-bundle over $B$. Thus, by the Proper Base Change Theorem in topology, the derived pushforward $R\rho_*\underline{\mathbb{Z}}$ is isomorphic to the pullback of $R\pi_*\underline{\mathbb{Z}}$. This is quasi-isomorphic to a two-term complex concentrated in degrees $0$ and $1$ whose cohomology sheaves are both isomorphic to the locally constant sheaf $\underline{\mathbb{Z}}$ on $B$. The terms in the induced Leray–Serre Spectral Sequence are naturally modules over the cohomology ring $H^*(B;\mathbb{Z})$. Thus, as above, the spectral sequence reduces to $$\dotsb \to H^{n-1}(B;\mathbb{Z}) \xrightarrow{c_1 \cup -} H^{n+1}(B;\mathbb{Z}) \to H^{n+1}(E;\mathbb{Z}) \to \dotsb. $$ Finally, for every Lie group $G$ (including a finite group with the discrete structure), for every continuous action of $G$ on a CW complex $X$, $$\mu:G\times X \to X,$$ for every morphism of Lie groups, $$\lambda:G\to \mathbb{S}^1,$$ set $B$ equal to $X\times^G EG$, and set $E$ equal to $(\mathbb{S}^1\times X)\times^G EG$, where the action of $G$ on $\mathbb{S}^1\times X$ is the diagonal action of $\lambda$ and $\mu$. In this case, the spectral sequence above reduces to $$\dotsb \to H_G^{n-1}(X;\mathbb{Z}) \xrightarrow{c_1 \cup -} H_G^{n+1}(X;\mathbb{Z}) \to H^{n+1}_G(\mathbb{S}^1\times X;\mathbb{Z}) \to \dotsb. $$

Answer 2

Like Andy's answer, this is about alternate proofs of the inequality in Theorem 1, which is, as was mentioned, perhaps due to Ed Floyd in a 1952 paper.

This Floyd theorem obviously implies the 1941 result of P. Smith: Smith Theorem: With $G$ and $X$ as in Theorem 1, if $X$ is mod $p$ acyclic, so is $X^G$.

In a paper that just appeared in Amer. J. Math., Chris Lloyd and I show that, curiously, this Smith theorem implies the Floyd theorem.

[One proves the contrapositive: if one had a $G$-space $X$ that contradicted the Floyd theorem, then one could find a $G$-space $Y$ that contradicted the Smith theorem. $Y$ is constructed as a retract of $X^n$ for a well chosen $n$, with the retraction coming from the representation theory of the $n$th symmetric group.]

So assuming Smith's 1941 result, this is a very different proof of Theorem 1.

[Our argument is general enough that it also applies with mod $p$ homology replaced by various Morava $K$-theories, and the real goal of our paper was to say something new about the Balmer spectrum of $G$-spaces for some nonabelian $2$-groups $G$.]

Answer 3

Since you also asked about alternate proofs of Theorem 1, there is a proof that does not use the localization theorem of a stronger version of the inequality (due to Floyd) in my notes Smith theory and Bredon homology. See Theorem B of those notes.