# Thread: Composition of homotopy equivalences is a homotopy equivalence.

1. ## 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?

2. ## Re: Composition of homotopy equivalences is a homotopy equivalence.

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 $f\simeq g$ and $g\simeq h$ implies $f\simeq h$), 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?