Modern geometry - methods and applications. Part III. Introduction to homology theory. Transl. from the Russian by Robert G. Burns. (English) Zbl 0703.55001

Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P.

Graduate Texts in Mathematics, 124. New York etc.: Springer-Verlag. ix, 416 p. DM 138.00 (1990).

This is a translation of the third part of the Russian original (1979; Zbl 0433.53001). It contains the following chapters: Homology and cohomoloy. Computational recipes; Critical points of smooth functions and homology theory; Cobordisms and smooth structures; Bibliography; Appendix 1 (by S. P. Novikov): An analogue of Morse theory for many-valued functions. Certain properties of Poisson brackets; Appendix 2 (by A. T. Fomenko): Plateau’s problem. Spectral bordisms and globally minimal surfaces in Riemannian manifolds; Index; Errata to Parts I and II.

MSC:

55-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
57Nxx	Topological manifolds
57Rxx	Differential topology
55M30	Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55N10	Singular homology and cohomology theory
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
70Hxx	Hamiltonian and Lagrangian mechanics
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

simplicial homology; Čech cohomology; homology theory; de Rham cohomology; singular homology; cohomology with local coefficients; cohomology rings of H-spaces; Lie groups; homology of bundles; characteristic classes; cohomology operations; rational homotopy groups; Poincaré duality; Lyusternik-Shnirel’man category; calculus of variations; geodesics; Bott periodicity; n-body problem in the plane; smooth cobordism; signature; representability of homology classes; rational Pontryagin classes; h-cobordism groups; smooth structures on manifolds; Critical points; Morse theory; Poisson brackets; Plateau’s problem; Spectral bordisms; globally minimal surfaces