2014/05/22 by Bhaskar Bagchi, Bagchi, Bhaskar · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.1405.5675
In a recent work [2] with Datta, we introduced the mu vector (with respect to\na given field) of simplicial complexes and used it to study tightness and lower\nbounds. In this paper, we modify the definition of mu vectors. With the new\ndefinition, most results of [2] become correct without the hypothesis of\n2-neighbourliness. In particular, the combinatorial Morse inequalities of [2]\nare now true of all simplicial complexes.\n As an application, we prove the following generalized lower bound theorem\n(GLBT) for connected locally tame combinatorial manifolds. If M is such a\nmanifold of dimension d, then for 1 \≤ \ℓ \≤ \(d-1)/(2) and any\nfield mathbbF, ~ g\ℓ+1 (M) \≥ binomd+2\ℓ+1\n\∑\i=1^\ℓ (-1)\ℓ-i \βi (M; mathbbF). Equality holds\nhere if and only if M is \ℓ-stacked.\n We conjecture that, more generally, this theorem is true of all triangulated\nconnected and closed homology manifolds. A conjecture on the sigma vectors of\ntriangulated homology spheres is proposed, whose validity will imply this GLB\nConjecture for homology manifolds. We also prove the GLBC for all connected and\nclosed combinatorial 3-manifolds. Thus, any connected closed combinatorial\nmanifold M of dimension three satisfies g2 (M) \≥ 10 \β1\n(M; mathbbF), with equality iff M is 1-stacked. This result settles a\nquestion of Novik and Swartz [6] in the affirmative.\n