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Math Help - Composition of homotopy equivalences is a homotopy equivalence.

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    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?
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    MHF Contributor Drexel28's Avatar
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    Re: Composition of homotopy equivalences is a homotopy equivalence.

    Quote Originally Posted by gummy_ratz View Post
    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?
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