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