Subject: Gaucher's question
From: Tom Goodwillie
Date: Thu, 6 Jul 2006 14:18:06 -0400
>
> > Subject: weak equivalence pi_k(X)-->pi_k(Y) if k>n
> > From: Gaucher Philippe
>
>> Date: Thu, 22 Jun 2006 16:00:01 +0200
>>
>> Dear All
>>
>> Is there a known construction on the category of compactly generated
>> topological spaces, or on the category of simplicial sets of a model
category
>
> > structure such that X-->Y is a weak equivalence iff pi_k(X)-->pi_k
(Y) is
an
> > isomorphism for k > n (n fixed).
>
>>
>> (for k> S^n-->D^{n+1}).
>>
> > Thanks in advance. pg.
I believe that the following is one solution of the problem.
First, I am going to assume that you mean based spaces, and that when you
say "pi_k(X)-->pi_k(Y) is an isomorphism for k>n" you mean just for the
given base points in X and Y. This is important. If I try interpreting
your definition of weak equivalence in an unbased sense, then I suppose
"pi_k(X)-->pi_k(Y) is an isomorphism for k>n" is meant to hold for all
possible base points in X, but then we have a violation of the
two-out-of-three condition. (This pitfall is related to one that I never
tire of pointing out: people sometimes misstate the definition of weak
homotopy equivalence of spaces by saying something like "induces an
isomorphism of pi_k(X)-->pi_k(Y) for all basepoints in X and all k greater
than equal to zero". This is bad because it makes the empty space weakly
equivalent to every nonempty space.)
Now, recall the usual (Quillen) model structure on based spaces (compactly
generated if you prefer).
Your weak equivalences are precisely those morphisms which become weak
equivalences in the usual sense when the functor \Omega^{n+1} is applied.
Let's decide that likewise the fibrations in the new model structure will
be precisely those morphisms which become fibrations when the functor
\Omega^{n+1} is applied. As usual let's call a map a trivial fibration if
it is both a fibration and a weak equivalence.
Let's call a map a cofibration if it has the left lifting property w.r.t.
trivial fibrations. We have good candidates for "generating cofibrations",
namely those morphisms that result from applying the left adjoint
\Sigma^{n+1} of \Omega^{n+1} to the usual generating cofibrations. By an
adjointness argument, the trivial fibrations are precisely the maps having
right lifting w.r.t. these. A standard small object argument allows you to
factor any map as a trivial fibration composed with a cofibration built
out of these.
Let's call a map a trivial cofibration if it has the left lifting property
w.r.t. fibrations. We have good candidates for "generating trivial
cofibrations", namely those morphisms that result from applying the left
adjoint \Sigma^{n+1} of \Omega^{n+1} to the usual generating trivial
cofibrations. By an adjointness argument, the fibrations are precisely the
maps having right lifting w.r.t. these. A standard small object argument
allows you to factor any map as a fibration composed with a trivial
cofibration built out of these.
It remains to verify that {trivial cofibrations} equals {cofibrations}
intersected with {weak equivalences}.
One inclusion follows from (every trivial fibration is a fibration) plus
(every trivial cofibration is a trivial cofibration in the usual sense).
The other follows from a standard retraction argument. (If f is a
cofibration and a weak equivalence, then write f=pi, where i is a trivial
cofibration and p is a fibration. p is a weak equivalence by 2 out of 3,
so f has the left lifting property w.r.t. p, so f is a retract of i, so f
is a trivial cofibration.)
Tom Goodwillie
P.S. Alternatively, I believe that you can use the usual fibrations. I
suppose that this would be called an example of colocalization. Generating
cofibrations can then be the usual generating trivial cofibrations plus
the based inclusions S^{k-1}-->D^k for k>n+1 and *-->S^{n+1}. You have to
verify that the trivial fibrations are precisely the maps having right
lifting w.r.t. these. The rest of the proof is then much as above.