Discrete morse theory and the cohomology ring

Forman, Robin

doi:10.1090/S0002-9947-02-03041-6

Discrete morse theory and the cohomology ring
HTML articles powered by AMS MathViewer

by Robin Forman PDF

Trans. Amer. Math. Soc. 354 (2002), 5063-5085 Request permission

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.