Subject: Re: 2 questions re htpy th Date: Fri, 2 May 2003 11:19:50 -0400 (EDT) From: Tom Goodwillie To: dmd1@lehigh.edu (Don Davis) > > A long time ago Sugawara proved that if T is a connected H space of the > homotopy type of a CW complex, there is a fibering of the form: > T-----> T*T ------>ST > where the projection is the Hopf construction on the multiplication. > Continuing the fiber sequence to the left, we have a map: > > LST ----> T > > where LST is the loop space of the suspension of T. Presumably this is > an extension of the multiplication: > > T x T ----> LST ----> T > > Does anyone know of a reference for this or a proof? Since there are > other maps T x T ----> T > for connected CW H spaces involving inverse maps, it is conceivable > that this presumption is false. I'm not sure which map T x T -----> LST you have in mind, but let's ask first what is the composition T -----> LST ----> T ? where T --> LST is the usual thing (unit of adjunction for L and S), not depending on the H-space structure. The obvious guess is the identity, in which case the composition T x T ---> T ---> LST ---> T would be the multiplication. Now of course the answer to any of these questions depends on which homotopy equivalence from T to the homotopy fiber of (T*T ---> ST) is being used; there is at least a little ambiguity because you can make at least one nontrivial equivalence from T to itself when T is a connected H-space. But let's assume things are set up in such a way that in the special case when T = LX for some connected based space X then we're talking about part of the fiber sequence ... ----> LSLX ----> LX ----> LX*LX ----> SLX ----> X So the map from LST to T is in that case obtained by applying the functor L to the map SLX ---> X that is the counit of the adjunction. So the composed map T ---> LST ---> T is in that case the identity LX ---> LSLX ---> LX. It's hard to imagine a natural map T ---> T for all connected H-spaces such that when T is LX it's homotopic to the identity but in general it's not.