Subject: Re: cofibrant G-space
Date: Tue, 12 Aug 2003 17:01:53 -0500
From: Bill Richter <richter@math.northwestern.edu>
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.