Johns Hopkins University

Classical homotopy theory has been well-developped both in the category of topological spaces and of

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)
and
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.