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Let $M^3$ be a homology sphere: a connected closed 3-manifold with the same homology groups as $S^3$. Calculate the first fundamental group and homology groups of the suspension $\Sigma M$. Use this to show that the suspension is homotopy-equivalent to $S^4$.

My Thought
I have shown that the fundamental group of the suspension is trivial and it has the same homology groups as $S^4$ but have trouble with proving it is homotopy equivalent to $S^4$. I tried to use the classification of homotopy types of four manifolds, as stated in this Wikipedia page https://en.wikipedia.org/wiki/4-manifold. However, it requires that $\Sigma M$ is a manifold, which is not clear to me.

Is it true that $\Sigma M$ is indeed a manifold in this case? If not, how to prove they are homotopy equivalent?

jlidm
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  • Are you familiar with the homology Whitehead theorem? – Connor Malin May 19 '22 at 17:04
  • @ConnorMalin Yes, but how can I construct a map from the suspension space to 4 sphere which induces isomorphisms on all homotopy groups ? – jlidm May 19 '22 at 17:08
  • Use the Hurewicz theorem. – Vincent Boelens May 19 '22 at 17:20
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    @VincentBoelens is correct that you can use Hurweicz theorem, but in this specific case you can use the fact that all orientable manifolds have a degree 1 map to the sphere, given by collapsing everything outside a ball. – Connor Malin May 19 '22 at 17:25
  • @ConnorMalin But we don't know whether or not the suspension space is a manifold in this case. – jlidm May 19 '22 at 17:31
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    But you know that $M$ is (and by the way $\Sigma M$ is a manifold, if and only if, $M$ is $S^n$). – Connor Malin May 19 '22 at 17:31
  • So firstly construct a degree 1 map from $M$ to $S^3$ and then it induces a map from the suspension space to $S^4$, which is a homotopy equivalence by Whitehead theorem? – jlidm May 19 '22 at 17:42
  • @ConnorMalin Where can I find a reference on your second statement about $\Sigma M$ ? – jlidm May 19 '22 at 17:44
  • https://math.stackexchange.com/questions/784962/easier-proof-about-suspension-of-a-manifold?rq=1 – Connor Malin May 20 '22 at 00:37

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