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Transactions of the American Mathematical Society

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Witten-Helffer-Sjöstrand theory
for $S^1$-equivariant cohomology


Author: Hon-kit Wai
Journal: Trans. Amer. Math. Soc. 351 (1999), 2141-2182
MSC (1991): Primary 58C40; Secondary 58F09
Published electronically: February 24, 1999
MathSciNet review: 1370653
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Abstract: Given an $S^1$-invariant Morse function $f$ and an $S^1$-invariant Riemannian metric $g$, a family of finite dimensional subcomplexes $(\widetilde \Omega^*_{inv,sm}(M,t), \break D(t))$, $t\in [0,\infty)$, of the Witten deformation of the $S^1$-equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian $\widetilde \Delta ^k(t)=D^*_k(t)D_k(t)+D_{k-1}(t)D^*_{k-1}(t)$ as $t\to \infty$. In fact the spectrum of $\widetilde \Delta^k(t)$ can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains $( \widetilde \Omega^*_{inv,sm}(M,t),D(t))$ as the complex of eigenforms corresponding to the small eigenvalues of $\widetilde \Delta(t)$. This permits us to verify the $S^1$-equivariant Morse inequalities. Moreover suppose $f$ is self-indexing and $(f,g)$ satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as $t\to \infty$ to a geometric complex which is induced by $(f,g)$ and calculates the $S^1$-equivariant cohomology of $M$.


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Additional Information

Hon-kit Wai
Affiliation: Department of Mathematics/C1200, University of Texas, Austin, Texas 78712
Address at time of publication: 4, 19/F, Nga Wo House, 50 Chun Wah Rd., Hong Kong

DOI: http://dx.doi.org/10.1090/S0002-9947-99-01711-0
Keywords: Schr\"odinger operators, equivariant Morse theory
Received by editor(s): October 24, 1995
Published electronically: February 24, 1999
Article copyright: © Copyright 1999 American Mathematical Society