## 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 C_{j} of f show immediately that MB_{t}(f)=P_{t}(M)+(1+t)R(t), where MB_{t}(f) is the MorseBott polynomial of f and P_{t}(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 f_{j}. This leads to a relationship between the kernels of the MorseSmaleWitten boundary operators associated to the Morse functions f _{j}and the perturbation of f. This method works when M and all the critical submanifolds are oriented or when ℤ2 coefficients areused.

Original language | English (US) |
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Pages (from-to) | 1693-1703 |

Number of pages | 11 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 29 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2009 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics