From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Loop spaces
Date: 16 Sep 1998 04:19:59 GMT
In article <6tmq40$rol$1@vixen.cso.uiuc.edu>, PC wrote:
>Anyone has seen a natural map (or even an embedding) of
>a pointed topological space X into its loop space? I know that
>we have one if X is a suspension of a connected space (by
>Milnor's splitting). What about in general?
Well, there are the constant maps X-> L(X) and of course maps homotopic
to those, but I imagine for ordinary spaces X you can't find any map
more exotic than that; embeddings are going to be hard to come by.
If P(X) is the path space of X, then the map P(X)->X, sending each
path to its endpoint, is a fibration with fibre L(X); since P(X) is
contractible, the homotopy long exact sequence shows pi_n(L(X)) is
naturally isomorphic to pi_{n-1}(X). So for example if X is any space
with Hom( pi_n(X), pi_{n-1}(X) ) = 0 for all n (e.g. if X is a
K(pi,n) for some n and pi) then all maps X -> L(X) induce the same
map in homotopy as a constant map, and so are themselves close to
null-homotopic (assuming X is taken from a category in which homotopy
groups detect all maps but "phantom maps").
Of course simple spaces (e.g. manifolds) usually have plenty of nonzero
homotopy groups, so at the algebraic level more maps X -> L(X) are
possible, but I suppose you'd have to construct them using a Postnikov-tower
argument. (I don't remember the details of Milnor's map, but I guess the
reason it succeeds when X = Sigma(Y) is that for such spaces X the
ring structure of H*(X) is trivial, so that there are really no
obstructions to constructing the map between X and L(X) one level at a
time in a Postnikov tower).
dave
("Obviously a bit rusty here")