Subject: wedges of spheres
Date: Tue, 27 May 2003 17:26:26 -0400
From: Tom Goodwillie
To: Don Davis
>
>
>> If X and Y are topological spaces, not homotopy equivalent to
>> singletons,
>> such that X wedge Y
>> has the homotopy type of a wedge of spheres, does it follow that X and
>Y
>> are
> > homotopy equivalent to wedges of spheres ??
I guess it's not hard to see in the simply connected case:
Prop: If X is a retract, in the homotopy category, of a wedge of
spheres of dimension >1, then X is homotopy equivalent to a wedge of
spheres.
Pf: A wedge of spheres has the following properties for every n
(1) the Hurewicz map from \pi_n(X) to (reduced integral) H_n(X) is onto
(2) H_n(X) is a free Z-module
and these properties are inherited by homotopy retracts.
On the other hand, if (1) and (2) are true then pick a Z-basis for
the homology, pick homotopy classes that map to these homology
classes, use these to map a wedge of spheres to X in such a way that
the induced map on homology takes the obvious basis of the homology
of the wedge of spheres to the chosen basis of the homology of X. So
this map induces an isomorphism in homology, and by the relative
Hurewicz theorem (using that X and the wedge of spheres are simply
connected) the map is a homotopy equivalence.
Tom Goodwillie