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Let $X$ be a topological space. Let $\Sigma$ be suspension.

Does $H^n(X;\mathbb{Z})\cong H^{n+1}(\Sigma X;\mathbb{Z})$ isomorphic or not?

Does $H^n(X;\mathbb{Z}_2)\cong H^{n+1}(\Sigma X;\mathbb{Z}_2)$ isomorphic or not?

Does $H^n(X;\mathbb{Z}_p)\cong H^{n+1}(\Sigma X;\mathbb{Z}_p)$ isomorphic for odd prime $p$ or not?

I tried to use long exact sequence of cohomology induced from $S^1\vee X\to S^1\times X\to \Sigma X$ and apply Kunneth formula on $H^*(S^1\times X;\mathbb{Z}_2)$, but I cannot work out.

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    $\begingroup$ Consider the long exact sequence associated to the pair $(CX,X)$, where $CX$ is the cone on $X$. $\endgroup$
    – user98602
    Commented Nov 5, 2014 at 5:33
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    $\begingroup$ Cross post mathoverflow.net/questions/186243/cohomology-of-suspension $\endgroup$
    – Rasmus
    Commented Nov 5, 2014 at 7:38
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    $\begingroup$ Just use Mayer-Vietoris... Besides you switched $n$ and $n+1$. PS: please wait some time before posting to MO if you need to post to MO at all. This kind of problem isn't research-level mathematics (which is what MO is about). $\endgroup$ Commented Nov 5, 2014 at 10:10
  • $\begingroup$ Thanks. I already understand by Prof. Mike's comment :) $\endgroup$
    – Shiquan
    Commented Nov 5, 2014 at 11:18
  • $\begingroup$ Is the same also true for reduced cohomology? $\endgroup$ Commented Sep 19, 2023 at 18:27

1 Answer 1

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Here's the argument given in the comment as an answer. Coefficients should be understood to be $\Bbb Z$.

Consider the pair $(CX,X)$, where $CX = X \times [0,1]/\sim$ , where $(x,1) \sim (x',1)$ for all $x,' \in X$, and $X$ is considered as the subspace $X \times \{0\}$ of $CX$. Then $CX/X \cong \Sigma X$.

Now we have a long exact sequence in reduced homology $$\dots \to H_{n+1}(CX,X) \to H_n(X) \to H_n(CX) \to H_n(CX,X) \to \dots$$

Now $CX$ is contractible so has trivial homology, so by exactness $H_{n+1}(CX,X) \cong H_n(X)$. Because $X$ is a closed subset of $CX$, we have that $H_{n+1}(CX,X) \cong H_{n+1}(CX/X) \cong H_{n+1}(\Sigma X)$, hence $H_{n+1}(\Sigma X) \cong H_n(X)$. Finally, thanks to the universal coefficient theorem for cohomology (Theorem 3.2 in https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) we get the desired results.

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