Proof that homotopy equivalent maps induce equivalent homomorphisms on homology groups…

Question

I am reading Hatcher, theorem 2.10.

( http://www.math.cornell.edu/~hatcher/AT/ATpage.html page 112 )

I mostly understand the proof, but am having trouble verifying that (except for the two), the terms with $i = j$ cancel. Why should the restriction of the homotopy to the faces $\sigma [v_0 , \ldots \hat v_i w_i \ldots w_n]$ and $\sigma [v_0 , \ldots v_i \hat w_i \ldots w_n]$ be equal? That doesn't seem to make sense - since these are definitely different subsets of $X \times I$. Or maybe they're not - I don't know. I'm finding the decomposition of $\Delta^n \times I$ into simplices difficult to visualize (too many lines for me in dimension three - are any tricks for seeing this more clearly? Or maybe a good picture...).

Answer 1

It's a bit confusing the way it's written. Take $i=j=1$ in the first sum, which gives the $n$-simplex $[v_0,w_1,\ldots,w_n]$. Now take $i=j=0$ in the second sum. This also gives the simplex $[v_0,w_1,\ldots,w_n]$.

In general, the term corresponding to $i=j=k+1$ in the first sum will cancel with the term corresponding to $i=j=k$ in the second sum. This holds for $k=0,\ldots,n-1$, which is why there are two remaining terms that don't cancel ($i=j=0$ in the first sum and $i=j=n$ in the second sum).