In the exercise of chapter 5 problem 21 in Rotman, he asks if $X$ is a topological space with 5 path components what is $H_1(CX,X)$, where $CX$ is the cone of X ?
I am generalising it to $n$ components and giving it a try:
We have an exact sequence for reduced Homology groups
$...\rightarrow \tilde { H}_1(X)\rightarrow \tilde H_1(CX) \rightarrow H_1(CX,X) \rightarrow \tilde H_0(X) \rightarrow \tilde H_0(CX) \rightarrow H_0(CX,X) \rightarrow 0 $ Since, $CX $ is contractible $\tilde H_1(CX)= H_1(CX)=0$ And since $CX$ is path connected, $\tilde H_0(CX)=0$ and $\tilde H_0(X)= \mathbb {Z}^{\oplus n-1}$
So, from the exact sequence we conclude $H_1(CX,X)= \mathbb {Z}^{\oplus n-1}$ .
Am I correct?
Is there any alternative way to see it?