The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds
The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds
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mu 向量、莫尔斯不等式和局部驯服组合流形的广义下界定理
DOI：
10.1016/j.ejc.2015.04.007
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
B. Bagchi
中科院分区：
文献类型：
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作者：
B. Bagchi
在 Datta 最近的一项工作（Bagchi 和 Datta，2014）中，我们引入了单纯复形的 mu 向量（相对于给定域），并用它来研究紧密性和下界。在本文中，我们修改了 mu 向量的定义。通过新的定义，Bagchi 和 Datta (2014) 的大部分结果在没有 2-邻域假设的情况下变得正确。特别是，Bagchi 和 Datta (2014) 的组合莫尔斯不等式现在适用于所有单纯复形。作为一个应用，我们证明了以下用于连接局部驯服组合流形的广义下界定理（GLBT）。如果 M 是这样一个维度为 d 的流形，则对于 1≤ ℓ≤ d− 1 2 和任意域 F，g ℓ+ 1 (M)≥(d+ 2 ℓ+ 1)Σ i= 1 ℓ (− 1) ℓ− i β i (M; F)。当且仅当 M 是 ℓ 堆叠时，等式在此成立。我们推测，更一般地，该定理对于所有三角连通且封闭的同调流形都成立。提出了关于三角同调球的西格玛向量的猜想，其有效性将暗示同调流形的GLB猜想。我们还证明了所有连通且封闭的组合 3 流形的 GLB 猜想。因此，任何连通的三维闭合组合流形 M 满足 g 2 (M)≥ 10 β 1 (M; F)，且当且仅当 M 是 1 堆叠的。这一结果肯定地解决了 Novik 和 Swartz (2009) 的问题。
In a recent work (Bagchi and Datta, 2014) with Datta, we introduced the mu-vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu-vectors. With the new definition, most results of Bagchi and Datta (2014) become correct without the hypothesis of 2-neighbourliness. In particular, the combinatorial Morse inequalities of Bagchi and Datta (2014) are now true of all simplicial complexes. As an application, we prove the following generalized lower bound theorem (GLBT) for connected locally tame combinatorial manifolds. If M is such a manifold of dimension d, then for 1≤ ℓ≤ d− 1 2 and any field F, g ℓ+ 1 (M)≥(d+ 2 ℓ+ 1)∑ i= 1 ℓ (− 1) ℓ− i β i (M; F). Equality holds here if and only if M is ℓ-stacked. We conjecture that, more generally, this theorem is true of all triangulated connected and closed homology manifolds. A conjecture on the sigma-vectors of triangulated homology spheres is proposed, whose validity will imply this GLB Conjecture for homology manifolds. We also prove the GLB Conjecture for all connected and closed combinatorial 3-manifolds. Thus, any connected closed combinatorial manifold M of dimension three satisfies g 2 (M)≥ 10 β 1 (M; F), with equality iff M is 1-stacked. This result settles a question of Novik and Swartz (2009) in the affirmative.