Discrete morse theory and the cohomology ring

Author:
Robin Forman

Journal:
Trans. Amer. Math. Soc. **354** (2002), 5063-5085

MSC (2000):
Primary 57Q99; Secondary 58E05

DOI:
https://doi.org/10.1090/S0002-9947-02-03041-6

Published electronically:
June 10, 2002

MathSciNet review:
1926850

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Abstract: In [5], we presented a discrete Morse Theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen [2] and Fukaya [11], [12].

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Additional Information

**Robin Forman**

Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77251

Email:
forman@math.rice.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03041-6

Received by editor(s):
August 13, 2001

Received by editor(s) in revised form:
January 30, 2002

Published electronically:
June 10, 2002

Additional Notes:
This work was partially supported by the National Science Foundation

Article copyright:
© Copyright 2002
American Mathematical Society