Re: loop group question

The standard reference for your last request is Milnor's `Spaces having
the homotopy-type of a CW-complex' TAMS 90 (1959).

But I think a `by-hands' proof of your original problem is not too bad.
Both these spaces have the subspace {f:S^1 -> G: f(*)=1, f'(*)=0}, so
it amounts to showing that this subspace is a deformation-retract of
both. Call the above space X, and your two spaces Y and Z. I think
the idea is to write any element y of Y as a sum x+b where x is in X,
and b is a bump function with support in a neighbourhood of *, chosen
suitably this is a unique representation of y. You can do the same
thing y = z + b_1 + b_2, etc. If you suitably restrict the type of
bump-functions allowed, this representation would be unique. That says
that your space Y is homeomorphic to a cartesian product X x (the lie
algebra of G), and Z is a cartesian product X x (lie algebra of G)^2.
This argument uses some standard facts about the topology on smooth
function spaces that you can find in Hirsch's differential topology
text.

I hope that helps,

-ryan

John Baez wrote:
Does anyone know a reference for the fact that
given a compact Lie group G, the usual smooth based loop group

{f: S^1 -> 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

.

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