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Face enumeration on simplicial complexes

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Recent Trends in Combinatorics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

In this chapter we survey many exciting developments on the face numbers of simplicial complexes from the past two decades. We focus on simplicial complexes whose geometric realizations are (homology) manifolds, as well as manifolds with additional combinatorial structure such as balanced manifolds or flag manifolds. The discussed results range from the Upper Bound Theorem for manifolds to the balanced Generalized Lower Bound Theorem for balanced polytopes.

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Notes

  1. 1.

    We further refer to Stanley’s recent article [84] which gives his personal account of how he came to the proof of the Upper Bound Theorem.

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Acknowledgements

We are grateful to Bhaskar Bagchi, Basudeb Datta, Satoshi Murai, Eran Nevo, and Ed Swartz for many helpful conversations and comments on the previous version of this paper.

Novik’s research is partially supported by NSF grant DMS-1361423.

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Authors and Affiliations

  1. Department of Mathematics, Seattle University, 901 12th Avenue, Seattle, WA, 98122, USA

    Steven Klee

  2. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    Isabella Novik

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  1. Steven Klee
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Correspondence to Isabella Novik .

Editor information

Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Klee, S., Novik, I. (2016). Face enumeration on simplicial complexes. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_26

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