Mathematical Research Letters

Volume 24 (2017)

Number 1

A Thom–Smale–Witten theorem on manifolds with boundary

Pages: 119 – 151



Wen Lu (School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei Province, China)


Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut–Lebeau’s analytic localization techniques. We extend Bismut–Zhang’s approach relying on the deformed Hodge–de Rham operator to the boundary case.

Received 16 September 2013

Accepted 17 September 2015

Published 7 June 2017