  Our objective is to develop a stratified Morse theory with tangential
conditions. We define a continuous strata-wise smooth Morse function on an
abstract stratified space by using control conditions and radiality assumptions
on the gradient vector field. For critical points of a Morse function one can
show that the local unstable set is a manifold and the local stable set is
itself an abstract stratified space. We also give a normal form for the
gradient dynamics in the neighborhood of critical points. For a stratified
Morse pair which satisfies the generic Morse-Smale condition we can build the
Morse-Witten complex and show that its homology is equivalent to the singular
homology of the space.

Geometric Topology