Discrete Morse-Bott theory
Odette Sylvia Yaptieu Djeungue
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Submission date: 12. Jan. 2017
MSC-Numbers: 05E99, 57R19, 57R70, 37B30
Keywords and phrases: Morse-Bott theory, Conley theory, discrete Morse theory, Polyhedral complexes, Poincaré polynomial, Betti numbers
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We derive a discrete analogue of Morse-Bott theory on polyhedral complexes and use this discrete Morse-Bott function to do some Conley theory analysis. We consider a function assuming the same value on maximal collections of polytopes, where the union of the closures of the polytopes in each collections should be connected. We define our discrete Morse-Bott function by requiring that for each collection of polytopes, the discrete Morse conditions have to be valid for those polytopes that have faces or are faces of polytopes not contained in that collection. We get an analogue of the (polynomial) Morse-Bott inequalities, only involving the Poincaré polynomials of our reduced collections all considered to have index zero. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory. We use the reduced collections as our isolated invariant sets, and define their respective isolating neighborhoods and exit sets, and the Poincaré polynomial (Euler number) of our polyhedral complex is obtained by summing up the Poincaré polynomials (Euler numbers) of the index pairs.