2

I was wondering where exactly we are using Excision.

I read the relation between relative homology and quotient homology recently. It says something like it can be applied on good pairs. I am looking for examples of good pairs upon which I can apply this.

Is $(B^n, S^{n-1})$ a good pair? I think yes. Take the open set $U=\{x\in B^n: 1/2 < \|x\|\leq 1\} \subset B$ and I think this open set of $B$ strongly deformation retracts to $S^{n-1}$ and now I can apply the corollary to Excision Theorem which states for all good pairs $(X,A)$ we have

$$H_n(X,A)=H_n(X/A, A/A)$$ where $H_n(X/A, A/A)$ is the reduced Homology of $X/A$. Am I correct?

Where else can I put Excision, its corollary to use?

permutation_matrix
  • 1,356

1 Answers1

1

There is a theorem that states that any CW-pair $(X,A)$ is a good pair. Here are two examples where you can use this:

  1. Consider the torus $T$ with its subcomplex $S^1 \vee S^1$. Since $T/(S^1 \vee S^1) \simeq S^2$, you get $H_n(T,S^1 \vee S^1) \cong H_n(S^2,*) \cong \tilde{H}_k(S^2) \cong (0,0,\mathbb{Z},0,0,...)$.

  2. Consider the CW-pair $(\mathbb{R}P^k,\mathbb{R}P^{k-1})$. If you look at the cell structure, you'll see that $\mathbb{R}P^k/\mathbb{R}P^{k-1} \cong S^k$, and hence $H_n(\mathbb{R}P^k,\mathbb{R}P^{k-1}) \cong H_n(S^k,*) \cong \tilde{H}_n(S^k)$.

Here is another nice example for when $(X,A)$ is not a CW-pair, but just a good pair:

  1. Consider $(T,S^1 \times \{0\})$, i.e. a torus with some fixed $S^1$ lying inside $T$. Then Hatcher shows in example 0.8 that $T / (S^1 \times \{0\}) \simeq S^2 \vee S^1$, and hence $H_n(T,S^1 \times \{0\}) \cong \tilde{H}_n(S^2 \vee S^1) \cong (0,\mathbb{Z},\mathbb{Z},0,0,...)$.

I hope this helps!

jasnee
  • 2,324