Arkiv för Matematik

Some combinatorial properties of flag simplicial pseudomanifolds and spheres

Christos A. Athanasiadis

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Abstract

A simplicial complex Δ is called flag if all minimal nonfaces of Δ have at most two elements. The following are proved: First, if Δ is a flag simplicial pseudomanifold of dimension d−1, then the graph of Δ (i) is (2d−2)-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the d-dimensional cross-polytope. Second, the h-vector of a flag simplicial homology sphere Δ of dimension d−1 is minimized when Δ is the boundary complex of the d-dimensional cross-polytope.

Dedication

Dedicated to Anders Björner on the occasion of his sixtieth birthday.

Note

Supported by the 70/4/8755 ELKE Research Fund of the University of Athens.

Article information

Source
Ark. Mat., Volume 49, Number 1 (2011), 17-29.

Dates
Received: 6 April 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907125

Digital Object Identifier
doi:10.1007/s11512-009-0106-4

Mathematical Reviews number (MathSciNet)
MR2784255

Zentralblatt MATH identifier
1235.52020

Rights
2009 © Institut Mittag-Leffler

Citation

Athanasiadis, Christos A. Some combinatorial properties of flag simplicial pseudomanifolds and spheres. Ark. Mat. 49 (2011), no. 1, 17--29. doi:10.1007/s11512-009-0106-4. https://projecteuclid.org/euclid.afm/1485907125


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References

  • Athanasiadis, C. A., On the graph connectivity of skeleta of convex polytopes, Discrete Comput. Geom. 42 (2009), 155–165.
  • Barnette, D., Graph theorems for manifolds, Israel J. Math. 16 (1973), 62–72.
  • Björner, A., Topological methods, in Handbook of Combinatorics, pp. 1819–1872, North-Holland, Amsterdam, 1995.
  • Gal, S. R., Real root conjecture fails for five- and higher-dimensional spheres, Discrete Comput. Geom. 34 (2005), 269–284.
  • Grünbaum, B., On the facial structure of convex polytopes, Bull. Amer. Math. Soc. 71 (1965), 559–560.
  • Meshulam, R., Domination numbers and homology, J. Combin. Theory Ser. A 102 (2003), 321–330.
  • Nevo, E., Remarks on missing faces and generalized lower bounds on face numbers, Electron. J. Combin. 16 (2009), Research Paper 8, 11 pp.
  • Novik, I., Personal communication, February 4, 2009.
  • Novik, I. and Swartz, E., Face ring multiplicity via CM-connectivity sequences, Canad. J. Math. 61 (2009), 888–903.
  • 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, 2nd ed., Progress in Mathematics 41, Birkhäuser, Boston, 1996.
  • Wotzlaw, R. F., Incidence Graphs and Unneighborly Polytopes, Doctoral Dissertation, TU Berlin, 2009.
  • Ziegler, G. M., Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, New York, 1995.