Face numbers and the fundamental group

2016/06/08 by Murai, Satoshi, Novik, Isabella
#05E45 #13F55 #57M05 #57Q15 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1606.02550

Abstract

We resolve a conjecture of Kalai asserting that the g₂-number of any simplicial complex Δ that represents a connected normal pseudomanifold of dimension d≥ 3 is at least as large as d+2 \choose 2m(Δ), where m(Δ) denotes the minimum number of generators of the fundamental group of Δ. Furthermore, we prove that a weaker bound, h₂(Δ)≥ d+1 \choose 2m(Δ), applies to any d-dimensional pure simplicial poset Δ all of whose faces of co-dimension ≥ 2 have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset Ψ all of whose vertex links satisfy Serre's condition (S_r), we establish lower bounds on h₁(Ψ),…,h_r(Ψ) in terms of the μ-numbers introduced by Bagchi and Datta.

Related

• Face numbers of manifolds with boundary
• Face numbers of pseudomanifolds with isolated singularities
• Remarks on missing faces and generalized lower bounds on face numbers
• Vertex numbers of simplicial complexes with free abelian fundamental group
• Combinatorial Lefschetz theorems beyond positivity