Discrete morse theory and the cohomology ring
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- by Robin Forman PDF
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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].References
- Eric Babson, Anders Björner, Svante Linusson, John Shareshian, and Volkmar Welker, Complexes of not $i$-connected graphs, Topology 38 (1999), no. 2, 271–299. MR 1660341, DOI 10.1016/S0040-9383(98)00009-3
- Martin Betz and Ralph L. Cohen, Graph moduli spaces and cohomology operations, Turkish J. Math. 18 (1994), no. 1, 23–41. MR 1270436
- Manoj K. Chari, On discrete Morse functions and combinatorial decompositions, Discrete Math. 217 (2000), no. 1-3, 101–113 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766262, DOI 10.1016/S0012-365X(99)00258-7
- Robin Forman, A discrete Morse theory for cell complexes, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 112–125. MR 1358614
- Robin Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no. 1, 90–145. MR 1612391, DOI 10.1006/aima.1997.1650
- Robin Forman, Witten-Morse theory for cell complexes, Topology 37 (1998), no. 5, 945–979. MR 1650414, DOI 10.1016/S0040-9383(97)00071-2
- Robin Forman, Combinatorial vector fields and dynamical systems, Math. Z. 228 (1998), no. 4, 629–681. MR 1644432, DOI 10.1007/PL00004638
- Robin Forman, Combinatorial differential topology and geometry, New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97) Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 177–206. MR 1731817
- Robin Forman, Morse theory and evasiveness, Combinatorica 20 (2000), no. 4, 489–504. MR 1804822, DOI 10.1007/s004930070003
- —, Novikov-Morse theory for cell complexes, to appear in Internat. J. of Math.
- Kenji Fukaya, Morse homotopy, $A^\infty$-category, and Floer homologies, Proceedings of GARC Workshop on Geometry and Topology ’93 (Seoul, 1993) Lecture Notes Ser., vol. 18, Seoul Nat. Univ., Seoul, 1993, pp. 1–102. MR 1270931
- Kenji Fukaya, Morse homotopy and its quantization, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 409–440. MR 1470740, DOI 10.1090/amsip/002.1/23
- J. Jonsson, On the homology of some complexes of graphs, preprint, 1998.
- —, The decision tree method, preprint, 1999.
- Varghese Mathai and Stuart G. Yates, Discrete Morse theory and extended $L^2$ homology, J. Funct. Anal. 168 (1999), no. 1, 84–110. MR 1717847, DOI 10.1006/jfan.1999.3439
- J. Shareshian, Discrete Morse Theory for Complexes of $2$-Connected Graphs, Topology, 40 (2001), pp. 681-701.
Additional Information
- Robin Forman
- Affiliation: Department of Mathematics, Rice University, Houston, Texas 77251
- Email: forman@math.rice.edu
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1926850