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Let $p$ and $q$ be two points of $\mathbb{R}^n$ where let $n\geq 1$. Then

$$\dim H^k(\mathbb R^n - p - q) = \begin{cases}0, &\text{ if }k\text{ is not equal to }n-1,\\ 2,&\text{ if }k = n-1.\end{cases}$$

I'm trying to prove this, but I've thought of letting $S = \{p,q\}$ and using the fact that the open set of $\mathbb R^{n+1} - S\times \mathbb{R}^1$ of $\mathbb R^{n+1}$ is homologically trivial in all dimensions.

  • 1
    Is this a homework question?2012-04-26
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    No, but just some extra practice problem2012-04-26
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    Can someone please help me on this2012-04-26
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    I assume, by your title, that you are doing DR cohomology? If so, state this.2012-04-26
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    Yes, that is correct2012-04-26
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    What do you know about DeRham Cohomology?2012-04-26
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    Pretty much, it boiled down to understanding the Mayer Vietoris theorem, and homotopy equivalence theorem2012-04-26

1 Answers 1

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Use Mayer-Vietoris with $U = \mathbb R^n - \{p\}, V = \mathbb R^n - \{q\}$. Then $U \cap V = \mathbb R^n -\{p,q\}$ and $U \cup V = \mathbb R^n$. Then Mayer-Vietoris gives $$ H^k(\mathbb R^n) \to H^k(U) \oplus H^k(V) \to H^k(\mathbb R^n - \{p,q\})\to H^{k+1}(\mathbb R^n). $$ But $H^*(\mathbb R^n)$ is trivial so you get $H^*(\mathbb R^n - \{p,q\}) \simeq H^k(U) \oplus H^k(V)$. But $U$ and $V$ are homotopic to $S^{n-1}$, giving you the desired result.