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Kirwan-Novikov inequalities on a manifold with boundary

Authors: Maxim Braverman and Valentin Silantyev
Journal: Trans. Amer. Math. Soc. 358 (2006), 3329-3361
MSC (2000): Primary 57R70; Secondary 58A10
Published electronically: February 20, 2006
MathSciNet review: 2218978
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Abstract: We extend the Novikov Morse-type inequalities for closed 1-forms in 2 directions. First, we consider manifolds with boundary. Second, we allow a very degenerate structure of the critical set of the form, assuming only that the form is non-degenerated in the sense of Kirwan. In particular, we obtain a generalization of a result of Floer about the usual Morse inequalities on a manifold with boundary. We also obtain an equivariant version of our inequalities.

Our proof is based on an application of the Witten deformation technique. The main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end. This leads to a considerable simplification of the local analysis. In particular, we obtain a new analytic proof of the Morse-Bott inequalities on a closed manifold.

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Additional Information

Maxim Braverman
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Valentin Silantyev
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Keywords: Novikov inequalities, Kirwan inequalities, Morse-Bott inequalities, Witten deformation
Received by editor(s): April 9, 2004
Published electronically: February 20, 2006
Additional Notes: This research was partially supported by NSF grant DMS-0204421
Article copyright: © Copyright 2006 American Mathematical Society

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