Date: Sat, 27 Jan 2001 20:31:22 -0600
From: Bill Richter <richter@math.northwestern.edu>
Subject: Re: 3 quick responses

Don, I hope you and other Alg Top teachers noticed that the result is

if A subset X is an closed NDR pair,
then X cup CA ---> X/A is a h.e.

and NDR pair is implied by sub-CW complex.  Whitehead and Gray (and
surely May) prove the NDR pair result and make the deduction.

The real point is that there's a Strom model category on TOP (as Dwyer
& Spalinski point out), where

w.e.s = actual h.e.s,
cofibrations = NDR pairs.

Folks think they work in the Quillen model category on TOP where

w.e.s = w.h.e.s,
cofibrations = sub-CW complex or retracts thereof

but this is a very weak model category.  It's where we state our
results, but to prove them, we often have to use either the Strom
model category on TOP, or else simplicial sets.   Point is that there
are 2 extra axioms that are true in both Strom & simplicial sets:

Pulling back a fibration along a cofibration gives a cofibration of
total spaces.

If A -i--> B >-j--> X and A >-ji--> X, then A >-i--> B.

Neither one of these axioms are true in the Quillen model category on
TOP.  Both are obviously true in simplicial sets, since there
cofibration = levelwise monomorphism.  Whitehead proves the 1st axiom
in his book (it's Strom's proof word for word), but doesn't mention
the second axiom, which is proved in Strom's later paper cited below.
This second axiom is needed to make sense out of pointed categories
like *\TOP: If X has a good basepoint, then Omega X also has a good
basepoint.  This uses both axioms together with a result Strom proves
by hand, that

If A >---> X then A^I  >---> X^I

Again this is surely true for simplicial sets, but not the Quillen
model category on TOP.  This later Strom axiom is even more important
in fiberwise homotopy theory, where you have a "bigger" basepoint.



  author =       "A. {Str{\ooo}m}",
  title =        "The homotopy category is a homotopy category",
  journal =      "Arch. Math.",
  year =         1972,
  volume =       23,
  pages =        "435--441"


  author =       "W. Dwyer and J. Spalinski",
  title =        "Homotopy theory and model categories",
  note =         "James handbook",
  year =         1994