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Fibrewise CW-complexes and Fibrewise Morse Theory
Johns Hopkins University
Classical homotopy theory has been well-developped both
in the category of topological spaces and of G-spaces
(topological spaces with a continuous action by a topological group G).
In the category of interest to us, namely
that of spaces over a fixed base-space B,
whose objects are
one can also do homotopy theory.
This was first discovered 30 years ago by a
number of people independantly.
However, the theory is not nearly as
complete as in the other two contexts.
This is primarily due to the lack
of a good definition of CW-complex.
Recalling how classical Morse theory, via the use of Morse functions,
yields CW-complex structures on smooth manifolds,
we consider fibrewise
Morse functions for a submersion (or smooth fibre bundle)
study how such Morse functions yield geometric information on X.
This study, together with the categorical approach to homotopy
using Quillen's ``model categories",
suggests a tentative definition of fibrewise CW-complexes.
If a conjecture concerning these objects turns out to be true,
this will be the desired correct notion of fibrewise CW-complex.