×

A tightness criterion for homology manifolds with or without boundary. (English) Zbl 1322.57019

Let \({\mathbb F}\) be any field. A connected simplicial complex \(X\) is \({\mathbb F}\)-tight if, for every induced subcomplex \(Y\) of \(X\), the homology maps \(H_*(Y;\mathbb F) \to H_*(X;{\mathbb F})\) induced by inclusions are injective. In the present paper, the author generalizes to \({\mathbb F}\)-homology manifolds some criteria proving tightness.
An \({\mathbb F}\)-homology manifold of dimension \(d\) is a simplicial complex all whose vertex links are \({\mathbb F}\)-homology spheres or \({\mathbb F}\)-homology balls of dimension \(d-1\) (an \({\mathbb F}\)-homology sphere is a simplicial complex such that the link \(L\) of every face has the same \({\mathbb F}\)-homology as the sphere of dimension \(\text{dim}(L)\) and we refer to the paper for the inductive definition of \({\mathbb F}\)-homology balls).
The two main results of the paper assert \({\mathbb F}\)-tighness for \((k+1)\)-neighbourly \(k\)-stacked \({\mathbb F}\)-homology manifold with boundary (Theorem 13) or, with the additive assumption of \({\mathbb F}\)-orientability, without boundary (Theorem 15); a simplicial complex \(K\) is \(n\)-neighbourly if any subset of \(n\) vertices form a face of \(K\) and a \((d+1)\)-dimensional \({\mathbb F}\)-homology manifold \(K\) with boundary is \(k\)-stacked if \(K\) and its boundary \(\partial K\) have the same \((d-k)\)-skeleton.
The last result gives a complete answer to a question asked by F. Effenberger in [J. Comb. Theory, Ser. A 118, No. 6, 1843–1862 (2011; Zbl 1282.52015)].

MSC:

57Q15 Triangulating manifolds
52B70 Polyhedral manifolds
05E99 Algebraic combinatorics

Citations:

Zbl 1282.52015
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[2] Bagchi, B.; Datta, B., On \(k\)-stellated and \(k\)-stacked spheres, Discrete Math., 313, 2318-2329 (2013) · Zbl 1297.57057
[3] Bagchi, B.; Datta, B., On polytopal upper bound spheres, Mathematika, 50, 493-496 (2013) · Zbl 1272.52016
[4] Bagchi, B.; Datta, B., On stellated spheres and a tightness criterion for combinatorial manifolds, European J. Combin., 36, 294-313 (2014) · Zbl 1301.52032
[5] Banchoff, T. F., Tightly embedded 2-dimensional polyhedral manifolds, Amer. J. Math., 87, 462-472 (1965) · Zbl 0136.21005
[6] Banchoff, T. F., Tight polyhedral Klein bottles, projective planes and Möbius bands, Math. Ann., 207, 233-243 (1974) · Zbl 0275.57005
[7] Datta, B.; Singh, N., An infinite family of tight triangulations of manifolds, J. Combin. Theory Ser. A, 120, 2148-2163 (2013) · Zbl 1383.57028
[8] Effenberger, F., Stacked polytopes and tight triangulations of manifolds, J. Combin. Theory Ser. A, 118, 1843-1862 (2011) · Zbl 1282.52015
[9] Kühnel, W., Higher dimensional analogues of Csaszar’s torus, Results Math., 9, 95-106 (1986) · Zbl 0552.52003
[10] Kühnel, W., Tight polyhedral manifolds and tight triangulations, (Lecture Notes in Mathematics, vol. 1612 (1995), Springer Verlag: Springer Verlag Berlin) · Zbl 0834.53004
[11] Kühnel, W.; Lutz, F. H., A census of tight triangulations, Period. Math. Hungar., 39, 161-183 (1999) · Zbl 0996.52014
[12] Murai, S.; Nevo, E., On the generalized lower bound conjecture for polytopes and spheres, Acta Math., 210, 185-202 (2013) · Zbl 1279.52014
[13] Murai, S.; Nevo, E., On \(r\)-stacked triangulated manifolds, J. Algebraic Combin., 39, 373-388 (2014) · Zbl 1383.57029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.