2
$\begingroup$

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...).

$\endgroup$
2
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.