Subject: loop group question
From: John Baez
Date: Sun, 29 Oct 2006 13:32:05 -0800
Dear Topologists -
Does anyone know a reference for the fact that
given a compact Lie group G, the usual based loop group
{f: S1 -> G: f(*) = 1, f smooth}
is homotopy equivalent to this slightly different one:
{f: [0,1] -> G: f(0) = f(1) = 1, f smooth} ?
The first is a proper subgroup of the second, since we demand smoothness
at the basepoint *.
Here I'm giving both these groups their C^infinity topology.
I think I could prove this quickly if both these groups have the homotopy
type of a CW complex. So, a reference for that would also make me happy.
Best,
jb