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.
Quote:
Originally Posted by
gummy_ratz 
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?
Since you can follow the proof, I guess what you're looking for is some motivation. Recall the proof of the transitivity of the relation 'homotopic' (i.e. that 
and 
implies 
), remember how you went 'half time' on the first half, and you want them to be the 'second map' at the middle? There is a complete analogy here, do you see it?
