Stratified Morse theory

Stratified Morse theory
Name	Stratified Morse theory
Field	Algebraic topology; Differential topology; Singularity theory
Introduced	1970s–1980s
Founders	Mark Goresky; Robert MacPherson
Related	Morse theory; Intersection homology; Perverse sheaves

Stratified Morse theory is a framework that extends classical Morse theory to spaces with singularities, providing tools to study topology of stratified spaces via critical points of functions compatible with a given stratification. The theory connects ideas from Mark Goresky and Robert MacPherson with techniques influenced by work of René Thom, John Milnor, Raoul Bott, Lê Dũng Tráng, and Pierre Deligne, and plays a central role in the study of singular algebraic varieties, complex analytic spaces, and topological invariants used by researchers at institutions such as Institute for Advanced Study and Harvard University.

Introduction

Stratified Morse theory generalizes Morse's ideas about nondegenerate critical points on smooth manifolds—developed by Marston Morse and popularized through connections with John Milnor and Raoul Bott—to spaces decomposed into smooth strata, as in the work of René Thom and Hassler Whitney. It provides homological and cohomological information akin to classical Morse inequalities, informed by algebraic methods of Pierre Deligne and Alexander Grothendieck, and influenced by geometric representation theory from groups like École Normale Supérieure and universities such as Princeton University.

Background and Definitions

A stratified space in this setting is a topological space endowed with a filtration by closed subsets whose successive differences are smooth manifolds; this notion builds on stratifications introduced by Hassler Whitney and axiomatized by René Thom and later refined by researchers associated with Institut des Hautes Études Scientifiques and École Polytechnique. Key definitions include strata, control data, and conical neighborhood structures inspired by constructions from John Mather and techniques related to Lê Dũng Tráng's work on complex singularities. A stratified Morse function is a real-valued function compatible with the stratification, having nondegenerate behavior transverse to strata, a notion formalized by Mark Goresky and Robert MacPherson and influenced by ideas from Shing-Tung Yau and William Thurston.

Main Theorems and Results

Fundamental results include stratified versions of the Morse lemma, Morse inequalities, and handle attachment phenomena, originally proven in the foundational texts by Mark Goresky and Robert MacPherson and later extended by scholars at Massachusetts Institute of Technology and University of California, Berkeley. The main theorems relate critical points on strata to changes in intersection homology groups introduced by Mark Goresky and Robert MacPherson, with connections to Poincaré duality statements refined by Pierre Deligne and George Lusztig. Other major results interweave with the theory of perverse sheaves developed by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne, and with monodromy theorems studied by Lê Dũng Tráng and Jean Leray.

Examples and Applications

Applications appear across singularity theory, algebraic geometry, and representation theory, such as in the study of singular algebraic varieties investigated at University of Cambridge and University of Oxford, complex hypersurface singularities examined by John Milnor and Lê Dũng Tráng, and orbit stratifications for actions of Lie groups like Special Linear Group studied in geometric representation theory by researchers at Clay Mathematics Institute and MPI for Mathematics. Concrete examples include analysis of complex plane curve singularities, conical singularities on algebraic varieties studied by Alexander Grothendieck-inspired algebraic geometers, and applications to the topology of moduli spaces considered at Stanford University and University of Chicago.

Techniques and Proof Sketches

Techniques combine transversality arguments from René Thom and John Mather, sheaf-theoretic methods from Pierre Deligne and Alexander Beilinson, and local analytic control inspired by Lê Dũng Tráng and Hironaka; proofs often use induction on strata dimensions, control data to manage tubular neighborhoods, and variants of the Morse lemma adapted to conical models. Homological calculations employ intersection homology of Mark Goresky and Robert MacPherson, and categorical arguments use perverse sheaves as organized by Joseph Bernstein and Zoghman Mebkhout, with input from topologists at Princeton University and algebraic geometers at Institut des Hautes Études Scientifiques.

Relations to Other Theories

Stratified Morse theory interfaces with intersection homology, perverse sheaves, and microlocal analysis as developed by Masaki Kashiwara and Masaki Kashiwara-affiliated schools, and connects to Hodge theory influenced by Pierre Deligne and Wilfried Schmid. It also relates to the study of Morse–Bott theory stemming from Raoul Bott and to equivariant topology for group actions considered by Michael Atiyah and Isadore Singer, with downstream consequences in geometric representation theory explored by George Lusztig and David Kazhdan.

Historical Development and Key Contributors

The theory was formulated and developed primarily by Mark Goresky and Robert MacPherson in the late 1970s and early 1980s, building on foundational work by René Thom, Marston Morse, and John Milnor, and influenced by contemporary advances by Pierre Deligne, Raoul Bott, and Lê Dũng Tráng. Subsequent contributors include researchers from Massachusetts Institute of Technology, Harvard University, Princeton University, and Institut des Hautes Études Scientifiques, and later expansions involved scholars working on perverse sheaves and microlocal analysis at institutions such as Clay Mathematics Institute and Max Planck Institute for Mathematics.

Category:Algebraic topology