Calculate $H^*(M - \{ p \})$ in terms of $H^*$ for $M$ a closed manifold

Question

This is from my graduate level differential geometry class.

Let $M$ be a closed manifold. I am trying to calculate $H^*(M - \{ p \})$ in terms of $H^*$.

Here is what I have so far: We know from excision theorem that the cohomology of the punctured manifold $M - \{ p \}$ can be computed using the long exact sequence of $(M, M - \{ p \})$ given by

$$ \cdots \to H^n(M, M - \{ p \}) \to H^n(M) \to H^n(M - \{ p \}) \to H^{n+1}(M, M - \{ p \}) \to H^{n+1}(M, M - \{ p \}) \to \cdots $$

Since $M$ is a closed manifold and $\{ p \}$ is a single point, we have that $H^n(M, M - \{ p \})$ is isomorphic to the reduced cohomology $\tilde{H}^{n-1}(S^{n-1})$. Thus we have $$ \cdots \to \tilde{H}^{n-1}(S^{n-1}) \to H^n(M) \to H^n(M - \{ p \}) \to \tilde{H}^n(S^n) \to \cdots$$

From this sequence, we have that $H^n(M - \{ p \})$ is isomorphic to $H^n(M)$ for $n \neq m-1, m$ where $m$ is the dimension of $M$, and there is an exact sequence $$ 0 \to H^{m-1}(M) \to H^{m-1}(M - \{ p \}) \to \mathbb Z \to H^m(M) \to H^m(M - \{ p \}) \to 0.$$

This gives the cohomology of $M - \{p\}$ in terms of the cohomology of $M$.

I am wondering if what I have so far is correct. If you could let me know where I can improve on and/or fix my mistake if there is any, that would be great.

Answer 1

Apart from the small things mentioned in the comments and the fact that it makes no use of the assumption that $M$ is closed, the proof looks fine to me. However, concluding with a long exact sequence feels somewhat weak here, and in fact if you know a little more about the (co)homology of closed manifolds you can take the argument a little bit further:

Let's consider the analogous exact sequence in homology $$ 0 \to H_m(M \setminus \{p\}) \to H_m(M) \overset{r_*}{\to} \underbrace{H_m(M, M \setminus \{p\})}_{\cong \mathbb{Z}} \to H_{m - 1}(M \setminus \{p\}) \to H_{m - 1}(M) \to 0 $$ where $r\colon (M, \emptyset) \to (M, M \setminus \{p\})$ is the restriction map. By the restriction theorem (cf. Hatcher, Algebraic Topology, Theorem 3.26), we only have to distinguish two cases:

1. $M$ is orientable and $r_*$ is an isomorphism. In this case $H_m(M \setminus \{p\}) = 0$ and $H_{m - 1}(M \setminus \{p\}) \cong H_{m - 1}(M)$.
2. $M$ is not orientable and $r_*$ is trivial. In this case $H_m(M \setminus \{p\}) \cong H_m(M) = 0$ and we obtain a short exact sequence $0 \to \mathbb{Z} \to H_{m - 1}(M \setminus \{p\}) \to H_{m - 1}(M) \to 0$.

However, this short exact sequence is in a sense the best we can do: Consider $M = \mathbb{R}\mathrm{P}^2$ with its standard CW-structure obtained from attaching a 2-cell to $\mathbb{R}\mathrm{P}^1 = S^1$. Let $x \in M$ be a point in the interior of this 2-cell so that $M \setminus \{x\}$ deformation retracts onto $S^1$. We then obtain the sequence $$ 0 \to \mathbb{Z} \to \underbrace{H_1(S^1)}_{\mathbb{Z}} \to \underbrace{H_1(\mathbb{R}\mathrm{P}^2)}_{\mathbb{Z} / 2} \to 0 $$ which does not split.

I'll leave it to you to derive analogous statements in cohomology via the universal coefficient theorem (don't expect them to be quite as clean) :)