Vertex numbers of simplicial complexes with free abelian fundamental group

2021/09/24 by Frick, Florian, Superdock, Matt · 1 citation
#05E45 #20F05 #57M05 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.11952

Abstract

We show that the minimum number of vertices of a simplicial complex with fundamental group ℤⁿ is at most O(n) and at least Ω(n^3/4). For the upper bound, we use a result on orthogonal 1-factorizations of K_2n. For the lower bound, we use a fractional Sylvester-Gallai result. We also prove that any group presentation ⟨ S | R⟩ ≅ ℤⁿ whose relations are of the form g^ah^bi^c for g, h, i ∈ S has at least Ω(n^3/2) generators.

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• Fundamental groups of small simplicial complexes

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