The Morse-Bott inequalities via a dynamical systems approach

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Abstract

Let f:M be a MorseBott 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 MorseBott polynomial of f and Pt(M) is the Poincar 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 MorseSmaleWitten boundary operators associated to the Morse functions f jand the perturbation of f. This method works when M and all the critical submanifolds are oriented or when ℤ2 coefficients areused.

Original languageEnglish (US)
Pages (from-to)1693-1703
Number of pages11
JournalErgodic Theory and Dynamical Systems
Volume29
Issue number6
DOIs
StatePublished - Dec 1 2009

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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