Face numbers and the fundamental group
Satoshi Murai, Isabella Novik

TL;DR
This paper proves new lower bounds on face numbers of simplicial complexes related to their fundamental groups, generalizing previous results and establishing bounds for various classes of simplicial posets.
Contribution
It resolves Kalai's conjecture relating $g_2$-numbers to the fundamental group and extends bounds to broader classes of simplicial posets.
Findings
Proves $g_2$-number bound in terms of fundamental group for pseudomanifolds.
Establishes $h_2$-number bound for certain simplicial posets.
Provides lower bounds on $h_1, \,\ldots,\, h_r$ for relative simplicial posets with Serre's condition.
Abstract
We resolve a conjecture of Kalai asserting that the -number of any simplicial complex that represents a connected normal pseudomanifold of dimension is at least as large as , where denotes the minimum number of generators of the fundamental group of . Furthermore, we prove that a weaker bound, , applies to any -dimensional pure simplicial poset all of whose faces of co-dimension have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset all of whose vertex links satisfy Serre's condition , we establish lower bounds on in terms of the -numbers introduced by Bagchi and Datta.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Advanced Combinatorial Mathematics · Homotopy and Cohomology in Algebraic Topology
