Discrete morse theory and the cohomology ring
Author:
Robin Forman
Journal:
Trans. Amer. Math. Soc. 354 (2002), 50635085
MSC (2000):
Primary 57Q99; Secondary 58E05
Published electronically:
June 10, 2002
MathSciNet review:
1926850
Fulltext PDF Free Access
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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].
 1.
E. Babson, A. Björner, S. Linussion, J. Shareshian and V. Welker, Complexes of not Connected Graphs, Topology, 38 (1999), pp. 271299. MR 2000a:57001
 2.
M. Betz and R. Cohen, Graph moduli space and cohomology operations, Turk. J. of Math., 18 (1994), pp. 2341. MR 95i:58037
 3.
M. Chari, On Discrete Morse Functions and Combinatorial Decompositions, Formal Power Series and Algebraic Combinatorics (Vienna, 1997), Discrete Math., 217 (2000), pp.101113. MR 2001g:52016
 4.
R. Forman, A Discrete Morse Theory for Cell Complexes, in Geometry, Topology and Physics for Raoul Bott, S.T. Yau (ed.), International Press, 1995. MR 97g:57030
 5.
, Morse Theory for Cell Complexes, Adv. in Math., 134 (1998), pp. 90145. MR 99b:57050
 6.
, WittenMorse Theory for Cell Complexes, Topology, 37 (1998), pp. 945979. MR 99m:58046
 7.
, Combinatorial Vector Fields and Dynamical Systems, Math. Zeit., 228 (1998), pp. 629681. MR 99f:58165
 8.
, Combinatorial differential topology and geometry, in New Perspectives in Algebraic Combinatorics (Berkeley, CA. 199697), Math. Sci. Res. Inst. Publ. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 177206. MR 2000h:57041
 9.
, Morse Theory and Evasiveness, Combinatorica, 20 (2001), pp. 489504. MR 2001k:57006
 10.
, NovikovMorse theory for cell complexes, to appear in Internat. J. of Math.
 11.
K. Fukaya, Morse homotopy, category, and Floer homologies, in Proc. Garc. Workshop on Geometry and Topology '93, (Seoul, 1993), Lecture Note ed. H. J. Kim, Lecture Note Ser. 18, Seoul National University, pp. 1102. MR 95e:57053
 12.
, Morse homotopy and its quantization, in Geometric Topology (Athens, GA 1993), AMS/IP Stud. Adv. Math 21, Amer. Math. Soc., Providence RI, 1997, pp. 409440. MR 98i:57061
 13.
J. Jonsson, On the homology of some complexes of graphs, preprint, 1998.
 14.
, The decision tree method, preprint, 1999.
 15.
V. Mathai and S. G. Yates, Discrete Morse theory and extended homology, J. Funct. Anal. 168 (1999), 84110. MR 2000i:58038
 16.
J. Shareshian, Discrete Morse Theory for Complexes of Connected Graphs, Topology, 40 (2001), pp. 681701.
 1.
 E. Babson, A. Björner, S. Linussion, J. Shareshian and V. Welker, Complexes of not Connected Graphs, Topology, 38 (1999), pp. 271299. MR 2000a:57001
 2.
 M. Betz and R. Cohen, Graph moduli space and cohomology operations, Turk. J. of Math., 18 (1994), pp. 2341. MR 95i:58037
 3.
 M. Chari, On Discrete Morse Functions and Combinatorial Decompositions, Formal Power Series and Algebraic Combinatorics (Vienna, 1997), Discrete Math., 217 (2000), pp.101113. MR 2001g:52016
 4.
 R. Forman, A Discrete Morse Theory for Cell Complexes, in Geometry, Topology and Physics for Raoul Bott, S.T. Yau (ed.), International Press, 1995. MR 97g:57030
 5.
 , Morse Theory for Cell Complexes, Adv. in Math., 134 (1998), pp. 90145. MR 99b:57050
 6.
 , WittenMorse Theory for Cell Complexes, Topology, 37 (1998), pp. 945979. MR 99m:58046
 7.
 , Combinatorial Vector Fields and Dynamical Systems, Math. Zeit., 228 (1998), pp. 629681. MR 99f:58165
 8.
 , Combinatorial differential topology and geometry, in New Perspectives in Algebraic Combinatorics (Berkeley, CA. 199697), Math. Sci. Res. Inst. Publ. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 177206. MR 2000h:57041
 9.
 , Morse Theory and Evasiveness, Combinatorica, 20 (2001), pp. 489504. MR 2001k:57006
 10.
 , NovikovMorse theory for cell complexes, to appear in Internat. J. of Math.
 11.
 K. Fukaya, Morse homotopy, category, and Floer homologies, in Proc. Garc. Workshop on Geometry and Topology '93, (Seoul, 1993), Lecture Note ed. H. J. Kim, Lecture Note Ser. 18, Seoul National University, pp. 1102. MR 95e:57053
 12.
 , Morse homotopy and its quantization, in Geometric Topology (Athens, GA 1993), AMS/IP Stud. Adv. Math 21, Amer. Math. Soc., Providence RI, 1997, pp. 409440. MR 98i:57061
 13.
 J. Jonsson, On the homology of some complexes of graphs, preprint, 1998.
 14.
 , The decision tree method, preprint, 1999.
 15.
 V. Mathai and S. G. Yates, Discrete Morse theory and extended homology, J. Funct. Anal. 168 (1999), 84110. MR 2000i:58038
 16.
 J. Shareshian, Discrete Morse Theory for Complexes of Connected Graphs, Topology, 40 (2001), pp. 681701.
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Additional Information
Robin Forman
Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77251
Email:
forman@math.rice.edu
DOI:
http://dx.doi.org/10.1090/S0002994702030416
PII:
S 00029947(02)030416
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
