I am self studying algebraic topology through vick's homology theorey, but I don't know how to prove 2.3 proposition on page 38~39. (which the author stated only and said it is easy)
Here's the proposition:
If $f:S^{n-1}\rightarrow Y $ is continuous where Y is Hausdorff, then there is an exact sequence $H_{m}\left(S^{n-1}\right) \stackrel{f_{\ast}}{\rightarrow} H_{m}\left(Y\right) \stackrel{i_{\ast}}{\rightarrow} H_{m}\left(D^{n}\cup_{f}Y\right) \stackrel{\Delta}{\rightarrow} H_{m-1}\left(S^{n-1}\right) \rightarrow...\rightarrow H_{0}\left(S^{n-1}\right)\rightarrow $$H_{0}(Y)\oplus Z \rightarrow H_{0}\left(D^{n}\cup_{f}Y\right)$
($ D^{n}\cup_{f}Y$ denotes attaching n-dim close ball (n-cell) to Y via map $f$, i.e $(D^{n}\cup Y )/\sim$ , where ~ is the relation generate by $x\sim f(x)$ )
He defines the connecting homomorphism ($\Delta$) via short exact sequence, but he didn't state what that sequence is in this proposition.
He didn't state what $i$ is neither, but I think it means the conanical quotient map $i: D^{n}\cup Y \rightarrow D^{n}\cup_{f}Y $ since $H_{m}\left(D^{n}\cup Y \right)=H_{m}\left( Y \right)$ when $ m>0 $
It may be an application of Mayer-Vietoris sequence, but I can't find how to apply.
Any one please help.
Consider the space $X=D^n\cup_f Y$ and lets define $U=X\setminus \{p\}$ where $p$ is the center of the ball $D^n$, and let $V=B_{\epsilon}(p)$ be a small open ball around the point $p$. Note that $U\cap V$ is within the interior of the ball $D^n$ and so it is just a copy of $S^{n-1}\times I$ or just $S^{n-1}$ up to homotopy. We also note that $U$ deformation retracts onto an embedded copy of $Y$ at the 'boundary' of the ball, so let us write the Mayer-Vietoris sequence associated to this decomposition of $X$.
$$\cdots\to H_m(U\cap V)\to H_m(U)\oplus H_m(V) \to H_m(X) \to H_{m-1}(U\cap V)\to\cdots$$
which from the above reasoning (for large enough $m$ such that $H_m(V)=0$) becomes
$$\cdots\to H_m(S^{n-1})\to H_m(Y) \to H_m(D^n\cup_f Y) \to H_{m-1}(US^{n-1})\to\cdots.$$
The maps in this sequence are exactly the maps appearing in yours as they are induced by homotopic maps (for instance your map $i$ is the inclusion of $Y$ into the 'boundary' of the $n$-ball $D^n$.)