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Balanced complexes and complexes without large missing faces

Michael Goff Department of Mathematics, University of Washington Steven Klee Department of Mathematics, University of Washington Isabella Novik Department of Mathematics, University of Washington

Geometric Analysis and Geometric Topology mathscidoc:1701.15003

Arkiv for Matematik, 49, (2), 335-350, 2009.7
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($d$−$l$)-CM, while the$l$-skeleton of a flag piecewise linear ($d$−1)-sphere is 2($d$−$l$)-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.
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@inproceedings{michael2009balanced,
  title={Balanced complexes and complexes without large missing faces},
  author={Michael Goff, Steven Klee, and Isabella Novik},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203629974229999},
  booktitle={Arkiv for Matematik},
  volume={49},
  number={2},
  pages={335-350},
  year={2009},
}
Michael Goff, Steven Klee, and Isabella Novik. Balanced complexes and complexes without large missing faces. 2009. Vol. 49. In Arkiv for Matematik. pp.335-350. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203629974229999.
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