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

Published electronically:
June 10, 2002

MathSciNet review:
1926850

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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].

**1.**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**, 10.1016/S0040-9383(98)00009-3**2.**Martin Betz and Ralph L. Cohen,*Graph moduli spaces and cohomology operations*, Turkish J. Math.**18**(1994), no. 1, 23–41. MR**1270436****3.**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**, 10.1016/S0012-365X(99)00258-7**4.**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****5.**Robin Forman,*Morse theory for cell complexes*, Adv. Math.**134**(1998), no. 1, 90–145. MR**1612391**, 10.1006/aima.1997.1650**6.**Robin Forman,*Witten-Morse theory for cell complexes*, Topology**37**(1998), no. 5, 945–979. MR**1650414**, 10.1016/S0040-9383(97)00071-2**7.**Robin Forman,*Combinatorial vector fields and dynamical systems*, Math. Z.**228**(1998), no. 4, 629–681. MR**1644432**, 10.1007/PL00004638**8.**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****9.**Robin Forman,*Morse theory and evasiveness*, Combinatorica**20**(2000), no. 4, 489–504. MR**1804822**, 10.1007/s004930070003**10.**-,*Novikov-Morse theory for cell complexes*, to appear in Internat. J. of Math.**11.**Kenji Fukaya,*Morse homotopy, 𝐴^{∞}-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****12.**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****13.**J. Jonsson,*On the homology of some complexes of graphs*, preprint, 1998.**14.**-,*The decision tree method*, preprint, 1999.**15.**Varghese Mathai and Stuart G. Yates,*Discrete Morse theory and extended 𝐿² homology*, J. Funct. Anal.**168**(1999), no. 1, 84–110. MR**1717847**, 10.1006/jfan.1999.3439**16.**J. Shareshian,*Discrete Morse Theory for Complexes of**-Connected Graphs*, Topology,**40**(2001), pp. 681-701.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57Q99,
58E05

Retrieve articles in all journals with MSC (2000): 57Q99, 58E05

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