Composition of homotopy equivalences is a homotopy equivalence.
The answer to this is in Question 2 Part 1 -> http://jedidiah.stuff.gen.nz/math/Al...oblem_set2.pdf .
I'm just trying to understand it. I follow it for the most part, but I just have a few questions about why they're doing certain things.
- I understand that we must check t=1/2 to see if it's continuous, but why do we define I(z,t) such that I(t,1/2) = g(h(z)) for both? Similarly below, why do we define D(x,1/2) = f(e(x)) for both?
- Why is it significant that I(x,0) = doi, and I(x,1) = idy
D(x,0) = iod, and D(x,1) = idz ?
I can follow the steps, but I'm just not exactly sure why they define I and D like they do? I don't see the big picture. I know we want doi = 1dz and iod = 1dx by definition of a homotopy equivalence... but I'm not sure what these checks are for?
Re: Composition of homotopy equivalences is a homotopy equivalence.