Lusternik–Schnirelmann theory

Lusternik–Schnirelmann theory
Name	Lusternik–Schnirelmann theory
Field	Algebraic topology; Calculus of variations
Introduced	1930s
Founders	Lev Pontryagin, Ludwig Hopf, Lyudmila Lusternik, Lev Schnirelmann

Lusternik–Schnirelmann theory is a branch of algebraic topology and calculus of variations that provides lower bounds on the number of critical points of smooth functions on manifolds by combining topological invariants with variational methods. Developed in the 1930s by Lyudmila Lusternik and Lev Schnirelmann, the theory interacts with concepts from Morse theory, homology theory, cohomology ring, cup product, and fixed point results such as the Borsuk–Ulam theorem, producing tools for problems in dynamical systems, differential geometry, and mathematical physics.

History and development

The origins trace to work by Lyudmila Lusternik and Lev Schnirelmann in the late 1920s and early 1930s, contemporaneous with contributions by Marston Morse, Solomon Lefschetz, and Andrey Kolmogorov, who influenced the interaction between topology and variational calculus. Subsequent advances involved researchers including Raoul Bott, Henri Poincaré, Ludwig Hopf, John Milnor, and René Thom, while modern refinements drew on work by Mikhail Gromov, Victor Klee, Raŭf G. Gadzhiev, and Ilya Dynnikov. Major developments occurred through collaborations and seminars at institutions like Moscow State University, Princeton University, University of Göttingen, and École Normale Supérieure, with later expansions by researchers affiliated to Institute for Advanced Study, Courant Institute, and Steklov Institute of Mathematics.

Definitions and fundamental concepts

Foundational definitions were formalized using notions from manifold theory, homotopy theory, singular homology, and cohomology theory as elaborated by Hassler Whitney, Henri Cartan, Samuel Eilenberg, and Norman Steenrod. Central constructs include the category of an open cover in the sense influenced by Pavel Alexandrov and the index notions akin to indices used by Marston Morse and Atiyah–Bott. The framework depends on smooth functions on compact manifolds as studied in works associated with Élie Cartan, Gleason–Yamabe, and Nikolay Bogolyubov, and it employs critical set analysis related to techniques from S. Smale and Stephen Smale-inspired dynamical topology. Algebraic inputs use the cup product in cohomology developed by J. H. C. Whitehead and computations facilitated by spectral sequences originating in research by Jean Leray and Samuel Eilenberg.

Lusternik–Schnirelmann category and index

The central invariant, the category (LS-category), assigns to a topological space a positive integer first explored alongside invariants of Hermann Weyl and André Weil; its precise formulation uses coverings by contractible sets relative to inclusion maps influenced by Poincaré duality results of Poincaré and later homotopy theoretic perspectives from J. H. C. Whitehead and Sergei Novikov. The category relates to the cup-length in cohomology ring as in computations by Jean-Pierre Serre and Beno Eckmann, and the index notion parallels indices in fixed point theory studied by Lefschetz and Brouwer. Inequalities comparing category, cup-length, and Betti numbers echo classical results by Alexander Grothendieck and Élie Cartan while being applied in contexts influenced by Michael Atiyah and Isadore Singer.

Main theorems and results

Principal results assert that any smooth function on a compact manifold has at least as many critical points as the category of the manifold, a statement formulated by Lyudmila Lusternik and Lev Schnirelmann and refined by Marston Morse and Raoul Bott. The cup-length lower bound, connecting category to cohomology operations, builds on work by Jean-Pierre Serre, Henri Cartan, and Norman Steenrod. Equivariant extensions tie to the Borsuk–Ulam theorem as treated by Karol Borsuk and to index theories reminiscent of the Atiyah–Singer index theorem from Michael Atiyah and Isadore Singer. Comparative inequalities between LS-category and Morse inequalities were elaborated by John Milnor and René Thom, while critical point existence results for Hamiltonian systems connect with work by Vladimir Arnold, S. Novikov, and Andrei Floer on symplectic topology. Developments in Lusternik–Schnirelmann theory also influenced conjectures and theorems investigated by Mikhail Gromov, Yuri Eliashberg, and Paul Rabinowitz.

Applications and examples

Applications include multiplicity results for closed geodesics on Riemannian manifolds studied by Marston Morse and George Birkhoff, existence theorems for periodic orbits in Hamiltonian dynamics traced to Vladimir Arnold and Dmitri Anosov, and existence of multiple solutions for nonlinear elliptic problems investigated by Paul Rabinowitz and Lions Struwe. Computations of category for spheres, projective spaces, and complex Grassmannians use methods from Hassler Whitney, Élie Cartan, Ada Yonath, and Raoul Bott, with explicit examples drawing from work at Princeton University and University of Cambridge. Equivariant and parametrized variants have been applied in problems studied by Albrecht Dold, Tibor Radó, and Marcel Berger, while connections to modern symplectic topology surface in research by Kenji Fukaya, Paul Seidel, and Andrei Floer.

Proof techniques and methods

Proofs use minimax constructions inspired by variational principles developed by David Hilbert and Emmy Noether, critical point theory shaped by Marston Morse and Stephen Smale, and algebraic tools from Samuel Eilenberg and Norman Steenrod such as spectral sequences of Jean Leray. Deformation arguments leverage flows reminiscent of constructions by George Reeb and René Thom, while cohomological lower bounds exploit cup products as in work by Jean-Pierre Serre and Henri Cartan. Equivariant proofs adapt techniques from fixed point theory due to Lefschetz and Borsuk, and analytic regularity relies on elliptic theory contributions by Louis Nirenberg and Lars Hörmander.

Category:Topology