Betti numbers of complexes with highly connected links

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• 作者：Amir Abu-Fraiha ^{aameer2@gmail.com} ; Roy Meshulam¹ ; ^{meshulam@math.technion.ac.il}
• 关键词：Simplicial homology ; Betti numbers ; Hypertrees
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：147
• 期：Complete
• 页码：27-40
• 全文大小：360 K
• 卷排序：147

文摘

Let Δn−1(k) denote the k  -dimensional skeleton of the (n−1)(n−1)-simplex Δn−1Δn−1 and consider a complex Δn−1(k−1)⊂X⊂Δn−1(k). Let KK be a field and let 0≤ℓ<k0≤ℓ<k. It is shown that if H˜k−ℓ−2(lk(X,τ);K)=0 for all ℓ-dimensional faces τ of X thendim⁡H˜k−1(X;K)≤(n−1ℓ)(n−ℓ−2k−ℓ)(k+1ℓ+1) with equality iff lk(X,τ)lk(X,τ) is a (k−ℓ−1)(k−ℓ−1)-hypertree for all ℓ-dimensional simplices τ   of Δn−1Δn−1. Examples based on sum complexes show that the bound is asymptotically tight for all fixed k,ℓk,ℓ as n→∞n→∞.