For any Subspace $A$ of a Path-Connected Space $X$, we have $H_0(X, A)=0$.

Asked Sep 01 '15 at 18:44

Active Sep 02 '15 at 09:47

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I recently learnt about relative homologies and am wondering if the following is true:

Statement: Let $X$ be a path-connected topological space and $A$ be a non-empty subspace of $X$. Then $H_0(X, A)=0$.

My argument is this: We need to show that the map $\bar \partial:C_1(X)/C_1(A)\to C_0(X)/C_0(A)$ is surjective.

So let $x+C_0(A)$ be an arbitrary member of $C_0(X)/C_0(A)$. Let $a\in A$ be arbitrary. Since $X$ is path-connected, there is a path $\gamma:I\to X$ such that $\gamma(1)=x$ and $\gamma(0)=a$.

Now $\bar \partial(\gamma+C_1(A))=\gamma(1)-\gamma(0)+C_0(A)=x-a+C_0(A)$. But since $a\in A$, we have $x-a+C_0(A)=x+C_0(A)$. Thus $x+C_0(A)$ is in the image of $\bar\partial$, meaning $\bar\partial$ is surjective.

Is the statement correct? If not, then can someone please point out an error in the argument? Thanks.

edited Sep 02 '15 at 09:47

Stefan Hamcke

27,733

asked Sep 01 '15 at 18:44

caffeinemachine

18,510

1

Looks fine to me. If you like, you can also prove this statement using the long exact sequence of homology groups. – Amitai Yuval Sep 01 '15 at 19:07
@AmitaiYuval Thank you. Using your approach, we just need to observe that the map $i_*:H_0(A)\to H_0(X)\cong \mathbb Z$ is surjective. Isn't it? – caffeinemachine Sep 01 '15 at 19:14
Yes, exactly... – Amitai Yuval Sep 01 '15 at 19:18
@AmitaiYuval "Exactly". Nice. Thanks for the help. – caffeinemachine Sep 01 '15 at 19:21
possible duplicate of $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$ – Najib Idrissi Sep 02 '15 at 09:32
Of course it's not a perfect duplicate, the linked question is much more general. But I don't think it's necessary to have both questions on the site. – Najib Idrissi Sep 02 '15 at 09:33
I think there is one missing step in the proof: An arbitrary member in $C_0(X)/C_0(A)$ is $\sum m_x \sigma_x$ for some $x \in X$. But the same argument follows. Please correct me if I am wrong. – Bryan Shih Oct 29 '17 at 17:12

For any Subspace $A$ of a Path-Connected Space $X$, we have $H_0(X, A)=0$.

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