# Homework Help: Proof in Homotopy Theory

1. May 3, 2010

### CornMuffin

1. The problem statement, all variables and given/known data
Prove that if Y is contractible, then any two maps from X to Y are homotopic.

I feel like I have a very, very, very sloppy proof :(

2. Relevant equations

3. The attempt at a solution
Assume Y is contractible to a point y0 held fixed, then there is a map F:Y x [0,1] -> Y such that
(i) F(y,0)=y0, for all y in Y
(ii) F(y,1)=y, for all y in Y
(iii) F(y0,t)=y0, for all t in [0,1]

Let F(y,t) be the homotopy of Y to a point y0
Claim: any map f:X -> Y is homotopic to y0 by the homotopy ft(x)=F(f(x),t)
Proof of Claim: Since f(x)=y in Y, ft=F(y,t), where y is in Y and t is in [0,1],
And since Y is contractible, f:X -> Y is homotopic to y0

Therefore, any two maps from X to Y are homotopic.

2. May 4, 2010

### lanedance

I don't know much about homotopies but you reasoning seems ok, it seems pretty clear that any map f is homotopic to the constant to y0.

The only assumption I can see in there, maybe worth checking you can use, is that a homotopy of functions is an equivalence class, ie.
f ~ y0
g ~ y0
implies
f ~ g