Let $X$ be a topological space and $x \in X$ be such that $\{x\}$ is closed in $X$. Let $y \in \mathbb{R}^n$. I've read that $$ H_k(\mathbb{R}^n \times X, (\mathbb{R}^n \times X) - \{(y,x)\}) \cong H_{k - n}(X,X - \{x\}). $$ for every $k$. I have no clue how to prove that. Can someone help?


First let me say that I'm really no expert on algebraic topology, only because there still wasn't an answer up to now I thought it might be helpful to give my suggestion for a proof anyways: Let us first assume $X$ is a CW complex. Further I assume you are referring to real coefficients (though this doesn't matter too much for the following).

  • From e.g. Hatchers book "Algebraic Topology", section 3.B, we know that there is an exact sequence of relative homologies, where we set $A:=X\setminus\{x\}$ and $B:=\mathbb{R}^n\setminus \{y\}$: $$\tag{1}0\rightarrow\oplus_i(H_i(X,A)\otimes_{\mathbb{R}}H_{k-i}(\mathbb{R}^n,B))\overset{(*)}{\rightarrow }H_k(X\times\mathbb{R}^n,A\times \mathbb{R}^n\cup X\times B)\rightarrow\oplus_i\mathrm{Tor}_{\mathbb{R}}(H_i(X,A),H_{k-i-1}(\mathbb{R}^n,B))\rightarrow 0 $$

  • Now, we use the long exact homology sequence (cf. section 2.1 of Hatchers book) $$\ldots\rightarrow H_i(B)\rightarrow H_i(\mathbb{R}^n)\rightarrow H_i(\mathbb{R}^n,B)\rightarrow H_{i-1}(B)\rightarrow\ldots$$ to prove that $H_i(\mathbb{R}^n,B)$ is $\mathbb{R}$ if $i=n$ and $0$ else (where we use the fact that the homology groups are invariant under homotopy and $\mathbb{R}^n$ is homotopic to a point, $B=\mathbb{R}^n\setminus\{y\}$ is homotopic to a sphere $S^{n-1}$). In particular, the Tor term in the exact sequence (1) vanisches, i.e. the morphism ($*$) is an isomorphism, and the direct sum of the first factor collapses to $H_{k-n}(X,A)\otimes_{\mathbb{R}}\mathbb{R}\simeq H_{k-n}(X,A)$.

  • Observing that $A\times\mathbb{R}^n\cup X\times B=X\times \mathbb{R}^n\setminus(\{x\}\times\{y\})$ finishes the prove.

To get rid of the assumption that $X$ is a CW complex you might want to read the paragraph in Hatchers book about CW approximation (in section 4.1), but I really don't know about this, I just have the vague idea it might help (sorry if it doesn't)...


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