# Equivariant cohomology of fixed points using the localisation theorem

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?
• That looks like the Wang Sequence. Commented Jul 6 at 18:31
• Actually it is the Thom-Gysin Sequence. Commented Jul 6 at 20:59
• @JasonStarr so the Thom-Gysin sequence associated to the bundle $S^1\to X\times_G S^1\to X_G$? If yes, why is the action of $\pi_1X_G$ on $X\times_G S^1$ trivial so that we can apply the Thom-Gysin sequence? Commented Jul 7 at 9:50
• @JasonStarr Also, if we have this Thom-Gysin sequence, the map $H_G^r(X)\to H_G^{r+2}(X)$ is given by the cup-product $t\smile (-)$ with $t\in H^2_G(X)$. But Tom-Dieck says that this map is the cup-product with some $t\in H^2(BG)$ (page 198 in Transformation Groups). Commented Jul 7 at 9:54
• The $\mathbb{S}^1$-bundle over $X_G$ is pulled back from $BG$. So the cohomology class is the pullback of the “first Chern class” of that circle bundle over $BG$. Commented Jul 7 at 11:40

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 $$\dotsb \to H_G^{n-1}(X;\mathbb{Z}) \xrightarrow{c_1 \cup -} H_G^{n+1}(X;\mathbb{Z}) \to H^{n+1}_G(\mathbb{S}^1\times X;\mathbb{Z}) \to \dotsb.$$

• Thank you editor for the edits! Commented Jul 9 at 15:06

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 $$G$$.]

• Is the paper in question this one? arxiv.org/abs/2008.00330 Commented Jul 9 at 20:02
• @AndyPutman Yes. Commented Jul 10 at 21:32

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.