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The Witten complex and the degenerate Morse inequalities. (English) Zbl 0608.58038

E. Witten [ibid. 17, 661-692 (1982; Zbl 0499.53056)] presented an analytic proof of the nondegenerate Morse inequalities, and suggested a method of proof of the degenerate Morse inequalities of Bott when the function has non-degenerate critical submanifolds. In this paper, the author uses the complex introduced by Witten, and gives a rigorous probabilistic proof of these inequalities. The method is based on an extension of the heat equation method for the proof of the index theorem by the author [J. Funct. Anal. 57, 56-99 (1984; Zbl 0538.58033)]. However, in the proof of the Morse inequalities of Bott, the Mayer- Vietoris argument is used. This question has recently received a complete analytical solution by B. Helffer and J. Sj√∂strand [Commun. Partial Differ. Equations 10, 245-340 (1985; Zbl 0597.35024)] and ’A proof of the Bott inequalities’, to appear in a volume in the honor of M. Sato.
Reviewer: K.Chang

MSC:

58J10 Differential complexes
58J20 Index theory and related fixed-point theorems on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58J65 Diffusion processes and stochastic analysis on manifolds
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