This book explores the natural generalization of Morse theory to stratified
spaces. Applications are given, primarily to the topology of complex analytic
varieties. The main theorems are proven here for the first time, although they
are heirs to a long line of historical development (see Sect. 1.7 and Sect. 2.8
of the introduction and Sect. 1.0 and Sect. 2.0 of Part I).
The work presented here was first announced in 1980 [GM1]. The original
proofs were discouragingly complicated and technical. During the intervening
years we have developed methods which have greatly simplified the arguments,
while at the same time making them more geometric (see Chap. 4 and Chap. 5
of Part I).
Conversations with P. Deligne, R. Lazarsfeld, and especially W. Fulton
about potential applications to complex varieties were instrumental in persuad-
persuading us to take up the study of stratified Morse theory. We have also profited
from valuable conversations with G. Bartel, L. Kaup, P. Orlik, P. Schapira,
B. Teissier, R. Thorn, R. Thomason, K. Vilonen, H. Hamm, Le D.T., D. Mas-
sey, W. Pardon, and D. Trotman. We would like to thank these last five, and
particularly D. Trotman, for their very careful reading of portions of several
preliminary versions of this manuscript.