Morse theory indomitable. (English) Zbl 0685.58009

This paper is a beautiful survey of the last fifty year’s progress in Morse theory with milestones Thom, Smale, Witten and Floer. Starting with his own half-space approach which led him to the celebrated periodicity theorems, the author describes the birth of the Thom-Smale-Witten complex of descending cells of the function f: \(M\to {\mathbb{R}}\), which yields the cohomology of the manifold M. Witten’s harmonic oscillator approach is expressed, and an outlook to the infinite-dimensional case is given by reviewing the works made by Atiyah, the author, and Floer. There is a lot of stories added with people like Steenrod, Morse, Thom, Smale, Wilson, Witten, Atiyah, Sternberg, Guillemin or Mumford, interlard with the author’s indomitable humour.
Reviewer: V.Zoller


58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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[1] M. F. Atiyah andR. Bott, The Yang-Mills equations over Riemann surfaces,Phil. Trans. Roy. Soc., London, A308 (1982), 523–615. · Zbl 0509.14014
[2] M. F. Atiyah andR. Bott, The moment map and equivariant cohomology,Topology,21 (1) (1984), 1–28. · Zbl 0521.58025
[3] J. J. Duistermaat andG. J. Heckman, On the variation in the cohomology in the symplectic form of the reduced phase space,Invent. Math.,69 (1982), 259–268. · Zbl 0503.58015
[4] A. Floer, Morse theory for Lagrangian intersections,J. Diff. Geom.,28 (1988), 513–547. · Zbl 0674.57027
[5] B. Helfer, J. Sjöstrand, Points multiples en mécanique semiclassique IV, étude du complexe de Witten,Comm. Par. Diff. Equ.,10 (1985), 245–340. · Zbl 0597.35024
[6] S. Smale, Differentiable dynamical systems,Bull. Am. Math. Soc.,73 (1967), 747. · Zbl 0202.55202
[7] René Thom, Sur une partition en cellules associée à une fonction sur une variété,C.R. Acad. Sci. Paris,228 (1949), 661–692. · Zbl 0034.20802
[8] E. Witten, Supersymmetry and Morse theory,J. Diff. Geom.,17 (1982), 661–692. · Zbl 0499.53056
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