Hi, I'm working on some homology problems but I need help figuring out the induced map from a given map, say [itex]f:X\rightarrow Y[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

For example, compute [itex]H_* (\mathbb{R}, \mathbb{R}^n - p)[/itex] where [itex]p \in \mathbb{R}^n[/itex].

So for [itex]n=1[/itex], we have the long exact sequence

[itex]0 \rightarrow H_1(\mathbb{R}^n-p)=0 \rightarrow H_1(\mathbb{R}^n)=0 \rightarrow H_1(\mathbb{R}^n, \mathbb{R}^n-p)[/itex]

[itex] \rightarrow H_0(\mathbb{R}^n-p)=\mathbb{Z}^2 \rightarrow H_0(\mathbb{R}^n)=\mathbb{Z} \rightarrow H_0(\mathbb{R}^n, \mathbb{R}^n-p)\rightarrow 0.[/itex]

So I got [itex]H_1(\mathbb{R}^n, \mathbb{R}^n-p)=\mathbb{Z}[/itex] but what is [itex]H_0(\mathbb{R}^n, \mathbb{R}^n-p)[/itex]? Is [itex]H_0(\mathbb{R}^n, \mathbb{R}^n-p)=\mathbb{Z}[/itex] or [itex]0[/itex]? I first thought it was [itex]\mathbb{Z}[/itex] because it's path connected, but I'm not so sure anymore. It's because [itex]H_0 (\mathbb{R}^n-p)=\mathbb{Z}^2[/itex]. So since [itex](\mathbb{R}^n-p) \rightarrow \mathbb{R}^n[/itex] is an inclusion map, what is the induced map [itex]H_0 (\mathbb{R}^n-p) \rightarrow H_0(\mathbb{R}^n)[/itex]? Is it injective or surjective? If I have this one key information, then I'm sure I can deduce the relative homology groups.

Now for [itex]n = 2[/itex], we have

[itex]0 \rightarrow H_2(\mathbb{R}^n-p)=0 \rightarrow H_2(\mathbb{R}^n)=0 \rightarrow H_2(\mathbb{R}^n, \mathbb{R}^n-p)[/itex]

[itex]\rightarrow H_1(\mathbb{R}^n-p)=\mathbb{Z} \rightarrow H_1(\mathbb{R}^n)=0 \rightarrow H_1(\mathbb{R}^n, \mathbb{R}^n-p)[/itex]

[itex] \rightarrow H_0(\mathbb{R}^n-p)=\mathbb{Z} \rightarrow H_0(\mathbb{R}^n)=\mathbb{Z} \rightarrow H_0(\mathbb{R}^n, \mathbb{R}^n-p)\rightarrow 0.[/itex].

I know [itex]H_2(\mathbb{R}^2, \mathbb{R}^2-p)=\mathbb{Z}[/itex] but are [itex]H_1(\mathbb{R}^2, \mathbb{R}^2-p)[/itex] and [itex]H_0(\mathbb{R}^2, \mathbb{R}^2-p)[/itex] isomorphic to the integers as well? If so, why?

Again, I think if I understand the following: if [itex]f:A\rightarrow X[/itex] is an inclusion map, is it always true that the induced map must be surjective? Can it be injective as well?

Thank you so much for your help!

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# Homology maps

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