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.