Discrete Morse functions from lexicographic orders

Authors:
Eric Babson and Patricia Hersh

Journal:
Trans. Amer. Math. Soc. **357** (2005), 509-534

MSC (2000):
Primary 05E25; Secondary 05A17, 05A18, 55P15

DOI:
https://doi.org/10.1090/S0002-9947-04-03495-6

Published electronically:
September 2, 2004

MathSciNet review:
2095621

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with and from any lexicographic order on its maximal chains. Specifically, if we attach facets according to the lexicographic order on maximal chains, then each facet contributes at most one new face which is critical, and at most one Betti number changes; facets which do not change the homotopy type also do not contribute any critical faces. Dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets. As one application, the Möbius function may be computed as the alternating sum of Morse numbers.

The above construction enables us to prove that the poset of partitions of a set with repetition is homotopy equivalent to a wedge of spheres of top dimension when is a hook-shaped partition; it is likely that the proof may be extended to a larger class of and perhaps to all , despite a result of Ziegler (1986) which shows that is not always Cohen-Macaulay.

**1.**E. Babson, A. Björner, S. Linusson, J. Shareshian and V. Welker,*Complexes of not**-connected graphs*, Topology**38**(1999), 271-299. MR**1660341 (2000a:57001)****2.**E. Babson and D. Kozlov,*Group actions on posets*, To appear in J. Algebra.**3.**L. Billera and G. Hetyei,*Linear inequalities for flags in graded partially ordered sets*, J. Combin. Theory Ser. A**89**(2000), 77-104.MR**1736134 (2001m:52011)****4.**A. Bjorner,*Shellable and Cohen-Macaulay partially ordered sets*, Trans. Amer. Math. Soc.**260**(1980), no. 1, 159-183. MR**0570784 (81i:06001)****5.**A. Björner,*Posets, regular CW complexes and Bruhat order*, European J. Combin.**5**(1984), no. 1, 7-16.MR**0746039 (86e:06002)****6.**A. Björner, Topological Methods, in*Handbook of Combinatorics*(R. Graham, M. Grötschel and L. Lovasz, eds.), North-Holland, Amsterdam, 1993.MR**1373690 (96m:52012)****7.**A. Björner and M. Wachs,*Bruhat order of Coxeter groups and shellability*, Adv. in Math.**43**(1982), 87-100. MR**0644668 (83i:20043)****8.**A. Bjorner and M. Wachs,*On lexicographically shellable posets*, Trans. Amer. Math. Soc.**277**(1983), no. 1, 323-341. MR**0690055 (84f:06004)****9.**A. Björner and M. Wachs,*Nonpure shellable complexes and posets I*, Trans. Amer. Math. Soc.**348**(1996), 1299-1327. MR**1333388 (96i:06008)****10.**A. Björner and M. Wachs,*Nonpure shellable complexes and posets II*, Trans. Amer. Math. Soc.**349**(1997), 3945-3975. MR**1401765 (98b:06008)****11.**M. Chari,*On discrete Morse functions and combinatorial decompositions*, Discrete Math**217**(2000), no. 1-3, 101-113. MR**1766262 (2001g:52016)****12.**J. Folkman,*The homology groups of a lattice*, J. Math. Mech.**15**(1966), 631-636.MR**0188116 (32:5557)****13.**R. Forman,*Morse theory for cell complexes*, Adv. Math.**134**(1998), 90-145.MR**1612391 (99b:57050)****14.**P. Hersh,*Lexicographic shellability for balanced complexes*, J. Algebraic Combin.**17**(2003), no. 3, 225-254. MR**2001670 (2004f:05190)****15.**P. Hersh,*On optimizing discrete Morse functions*, Preprint 2003.**16.**P. Hersh and R. Kleinberg,*The refinement complex of the poset of partitions of a multiset*, In preparation.**17.**P. Hersh and V. Welker,*Gröbner basis degree bounds on*, To appear in Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Integer Points in Polyhedra.**18.**D. Kozlov,*General lexicographic shellability and orbit arrangements*, Ann. of Comb.**1**(1997), no. 1, 67-90. MR**1474801 (98h:52023)****19.**D. Kozlov,*Complexes of directed trees*, J. Combin. Theory Ser A**88**(1999), no. 1, 112-122. MR**1713484 (2000j:05036)****20.**G-C. Rota,*On the foundations of combinatorial theory I: Theory of Möbius functions*, Z. Wahrsch.**2**(1964), 340-368. MR**0174487 (30:4688)****21.**J. Shareshian,*On the shellability of the order complex of the subgroup lattice of a finite group*, Trans. Amer. Math. Soc.**353**(2001), no. 7, 2689-2703.MR**1828468 (2002k:06006)****22.**R. Stanley,*Supersolvable lattices*, Algebra Universalis**2**(1972), 197-217.MR**0309815 (46:8920)****23.**R. Stanley, Enumerative Combinatorics, vol. I. Wadsworth and Brooks/Cole, Pacific Grove, CA, 1986; second printing, Cambridge University Press, Cambridge/New York, 1997.MR**0847717 (87j:05003)****24.**R. Stanley, Combinatorics and Commutative Algebra, second ed., Birkhäuser, Boston, 1996.MR**1453579 (98h:05001)****25.**V. Welker,*Direct sum decompositions of matroids and exponential structures*, J. Combin. Theory Ser. B**63**(1995), no. 2, 222-244. MR**1320168 (96e:05042)****26.**G. Ziegler,*On the poset of partitions of an integer*, J. Combin. Theory Ser. A**42**(1986), no. 2, 215-222.MR**0847552 (87k:06009)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
05E25,
05A17,
05A18,
55P15

Retrieve articles in all journals with MSC (2000): 05E25, 05A17, 05A18, 55P15

Additional Information

**Eric Babson**

Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

Email:
babson@math.washington.edu

**Patricia Hersh**

Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

Address at time of publication:
The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720-5070

Email:
phersh@msri.org

DOI:
https://doi.org/10.1090/S0002-9947-04-03495-6

Keywords:
Discrete Morse theory,
poset,
order complex,
partition

Received by editor(s):
July 1, 2003

Published electronically:
September 2, 2004

Article copyright:
© Copyright 2004
by Eric Babson and Patricia Hersh