From: Dan Christensen <jdc@julian.uwo.ca>
Subject: Re: Contractible spaces
Date: 29 Jan 2001 21:25:29 -0500
Newsgroups: sci.math.research
Summary: Contractibility of spaces of paths

Tom Leinster <leinster@dpmms.cam.ac.REMOVETHISBIT.uk> writes:

> I'm looking for a proof of the following result:
> 
> Theorem: Let X be a contractible space and x,x' points of X.  Then the space
>          X(x,x') of paths from x to x' is also contractible.
> 
> Before you dismiss this as standard and trivial, let me point out why it
> might not be.  (That is, I hope there *is* an easy low-tech proof, but it
> might not be the proof you're currently thinking of...:-))  
> 
> Contractibility means - according to my usage here, at least - that X is
> contractible as a plain, unadorned topological space, *not* as a
> space-with-basepoint.  

If you are willing to assume that the point X is contractible to is
non-degenerate, then you can show that you get a *pointed* contracting
homotopy, and you're all set.  So I'm going to presume that you don't
want to assume that X is well-pointed, and I hope that the following
explicit, low-tech argument, doesn't use this assumption implicitly!

Let H : X x I --> X be a homotopy from the identity map on X to the
constant map to a point x in X.  Let y and z be two points in X.  I'll
show that the space X(y,z) of paths from y to z is contractible.

Let a be the path from y to x given by t |--> H(y,t) and
let b be the path from z to x given by t |--> H(z,t).
I'll give a homotopy G : X(y,z) x I --> X(y,z) from the identity 
map to the constant map with value the path b'a from y to z.

The homotopy G sends a path c from y to z and a time t to the
composite of the following three paths:

  the portion of the path a from y to a(t) = H(y,t);
  the homotoped path s |--> H(c(s),t) from H(y,t) to H(z,t);
  and the portion of the path b' from b(t) = H(z,t) to z.

These should be parametrized so t/2 units of time are spent on the
first and third segments, and the rest on the middle segment.  So when
t = 0, this is just the path c, and when t = 1, it is the composite
path b'a (no time is spent on the constant homotoped path).

Dan