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arXiv · 0907.1669

Balanced complexes and complexes without large missing faces

Michael Goff·Steven Klee·Isabella Novik
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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 $2(d-l)$-CM while the $l$-skeleton of a flag PL $(d-1)$-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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Michael Goff, Steven Klee, Isabella Novik. 2009-07-09. Balanced complexes and complexes without large missing faces. https://arxiv.org/abs/0907.1669

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