On the group
of self homotopy
equivalences of
a Product
Philip R. Heath
Abstract
In this talk we exhibit two interlocking sequences
in order to study
the group
of based homotopy classes of
based self homotopy equivalences of
a product
of topological spaces X and Y.
These sequences have complementary features,
and the interconnectedness facilitates computation. The sequences
give new results about
,
they also
unify, generalize, and in some cases correct,
existing results in the literature about this group.
New results include calculations on the group of self equivalences of a
products of suspensions (with applications to a product of Moore
spaces, including p localized spheres);
some progress in the computation of
non-simply-connected rank 2 H-spaces (posed earlier as an open problem);
and a universal
description of
for
as a semidirect
product.
This last situation includes an example
that is not a
semidirect product when decomposed in the only other way known.
Current applications are to the product of two spaces, but the
techniques are clearly applicable to more than two. However
generalizations are not entirely straight forward since some of the
applications depend heavily on the fact that the
product of two suspensions can be written as the mapping cone of
a generalized Whitehead product. This is not available for the
product of more than two suspensions.
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