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The Morse–Bott inequalities via a dynamical systems approach

Published online by Cambridge University Press:  12 March 2009

AUGUSTIN BANYAGA  and
DAVID E. HURTUBISE
AUGUSTIN BANYAGA
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (email: [email protected])
DAVID E. HURTUBISE
Affiliation:
Department of Mathematics and Statistics, Penn State Altoona, Altoona, PA 16601-3760, USA (email: [email protected])
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Abstract

Let f:M→ℝ be a Morse–Bott function on a compact smooth finite-dimensional manifold M. The polynomial Morse inequalities and an explicit perturbation of f defined using Morse functions fj on the critical submanifolds Cj of f show immediately that MBt(f)=Pt(M)+(1+t)R(t), where MBt(f) is the Morse–Bott polynomial of f and Pt(M) is the Poincaré polynomial of M. We prove that R(t) is a polynomial with non-negative integer coefficients by showing that the number of gradient flow lines of the perturbation of f between two critical points p,q∈Cj of relative index one coincides with the number of gradient flow lines between p and q of the Morse function fj. This leads to a relationship between the kernels of the Morse–Smale–Witten boundary operators associated to the Morse functions fj and the perturbation of f. This method works when M and all the critical submanifolds are oriented or when ℤ2 coefficients are used.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1693 - 1703
Copyright
Copyright © Cambridge University Press 2009

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