Subject: Re: cofibrant G-space
Date: Tue, 12 Aug 2003 17:01:53 -0500
From: Bill Richter
To: dmd1@lehigh.edu
John R. Klein responded to Philippe Gaucher:
If one is willing to work with a Quillen type notion of weak
equivalence: X -> Y is a weak equivalence if it's a weak homotopy
equivalence of underlying spaces, then G-spaces can be equipped
with a model structure where the cofibrant spaces are retracts of
free G-cellular spaces.
John, Philippe said he wants an analogue of Strom's model structure
(w.e. = h.e.), and not Quillen's (w.e. = w.h.e). I constructed a
$pi$-equivariant Strom model category in an old preprint:
> I would be interesting in any reference concerning a similar
> model structure for G-spaces. And particularly how cofibrant
> objects looks like ? Probably a cofibrant G-space is a G-space
> with G acting freely in some sense on the space.
Sounds good, but what I know is this (from an old NSF proposal):
James's book "Fiberwise Homotopy" defines an equivariant Str{\ooo}m
structure, and proves that a closed $\pi$-pair $(X,A)$ possesses the
$\pi$ homotopy extension property iff $(X,A)$ possesses a
$\pi$-Str{\ooo}m structure. With this simple and bold result of
James, it's clear how to proceed. One defines a model category
structure on $\pi$-spaces with fibrations, cofibration and weak
equivalences defined by $\pi$-HLP, closed $\pi$-pairs having the
$\pi$-HEP, and $\pi$-homotopy equivalences. We then define
equivariant Hopf invariants and James constructions and so forth.
Recall Dwyer and Kan's model category structure for $\pi$-spaces,
whose cofibrant objects are the CW-$\pi$ complexes. I think that's
the category John mentions. By combining the two model category
structures, we obtain the EHP results needed for high dimensional
knots and Poincar\'e surgery.