If $j$ is the inclusion $A\hookrightarrow X$ then $j_{*}:H_1(X)\to H_1(X,A)$ is surjective.

Question

I need to prove that:

Let $X$ be a path-connected topological space and $A\subset X$. Given $j:X\hookrightarrow (X,A), x\mapsto j(x)=x$ prove that $j_{*}:H_1(X)\to H_1(X,A)$ is surjective.

Idea: Show that the diagram $A\stackrel{i}{\hookrightarrow} X\stackrel{j}{\hookrightarrow}(X,A)$ induce the following exact sequence: $$0\to H_1(A)\to H_1(X)\to H_1(X,A)\to 0$$ Is there any theorem to affirm this fact?

Answer 1

This is not true in general. Take $A=\partial I$, $X=I$, where $I=[0,1]$.

If $A$ is path-connected, then it is true. It can be seen to follow from the reduced homology long exact sequence:

$$\cdots \to H_1(A) \to H_1(X) \to H_1(X,A) \to \widetilde{H_0}(A)=0 \to \widetilde{H_0}(X) \to \cdots$$