Arkiv för Matematik

  • Ark. Mat.
  • Volume 49, Number 2 (2011), 335-350.

Balanced complexes and complexes without large missing faces

Michael Goff, Steven Klee, and Isabella Novik

Full-text: Open access

Abstract

The face numbers of simplicial complexes without missing faces of dimension larger than i are studied. It is shown that among all such (d−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal f-vector; and moreover, among all such 2-Cohen–Macaulay (2-CM) complexes, the same sphere has the componentwise minimal h-vector. It is also verified that the l-skeleton of a flag (d−1)-dimensional 2-CM complex is 2(dl)-CM, while the l-skeleton of a flag piecewise linear (d−1)-sphere is 2(dl)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.

Note

Novik’s research was partially supported by an Alfred P. Sloan Research Fellowship and NSF grant DMS-0801152.

Article information

Source
Ark. Mat., Volume 49, Number 2 (2011), 335-350.

Dates
Received: 10 July 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907140

Digital Object Identifier
doi:10.1007/s11512-009-0119-z

Mathematical Reviews number (MathSciNet)
MR2826947

Zentralblatt MATH identifier
1256.05259

Rights
2010 © Institut Mittag-Leffler

Citation

Goff, Michael; Klee, Steven; Novik, Isabella. Balanced complexes and complexes without large missing faces. Ark. Mat. 49 (2011), no. 2, 335--350. doi:10.1007/s11512-009-0119-z. https://projecteuclid.org/euclid.afm/1485907140


Export citation

References

  • Athanasiadis, C. A., Some combinatorial properties of flag simplicial pseudo-manifolds and spheres, to appear in Ark. Mat.
  • Baclawski, K., Cohen–Macaulay connectivity and geometric lattices, European J. Combin. 3 (1982), 293–305.
  • Barnette, D., Graph theorems for manifolds, Israel J. Math. 16 (1973), 62–72.
  • Barnette, D., A proof of the lower bound conjecture for convex polytopes, Pacific J. Math. 46 (1973), 349–354.
  • Björner, A., Topological methods, in Handbook of Combinatorics 2, pp. 1819–1872, Elsevier, Amsterdam, 1995.
  • Bredon, G., Topology and Geometry, Graduate Texts in Mathematics 139, Springer, New York, 1993.
  • Daverman, R., Decompositions of Manifolds, Pure and Applied Mathematics 124, Academic Press, Orlando, FL, 1986.
  • Fløystad, G., Cohen–Macaulay cell complexes, in Algebraic and Geometric Combinatorics, Contemp. Math. 423, pp. 205–220, Amer. Math. Soc., Providence, RI, 2006.
  • Fogelsanger, A., The Generic Rigidity of Minimal Cycles, Ph.D. Thesis, Cornell University, Ithaca, NY, 1988.
  • Frohmader, A., Face vectors of flag complexes, Israel J. Math. 164 (2008), 153–164.
  • Kalai, G., Rigidity and the lower bound theorem I, Invent. Math. 88 (1987), 125–151.
  • Meshulam, R., Domination numbers and homology, J. Combin. Theory Ser. A 102 (2003), 321–330.
  • Nevo, E., Rigidity and the lower bound theorem for doubly Cohen–Macaulay complexes, Discrete Comput. Geom. 39 (2008), 411–418.
  • Nevo, E., Remarks on missing faces and generalized lower bounds on face numbers, Electron. J. Combin. 16 (2009/2010), Research Paper #8, 11 pp.
  • Stanley, R. P., Balanced Cohen–Macaulay complexes, Trans. Amer. Math. Soc. 249 (1979), 139–157.
  • Stanley, R. P., A monotonicity property of h-vectors and h*-vectors, European J. Combin. 14 (1993), 251–258.
  • Stanley, R. P., Combinatorics and Commutative Algebra, Progress in Mathematics 41, Birkhäuser, Boston, MA, 1996.
  • Ziegler, G. M., Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, New York, 1995.