2014/06/17 by Bhaskar Bagchi, Bagchi, Bhaskar
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT
paper · pdf · doi:10.48550/arxiv.1406.4299
arxiv created 2014/06/17 · arxiv updated 2014/06/18
A simplicial complex X is said to be tight with respect to a field \mathbbF if X is connected and, for every induced subcomplex Y of X, the linear map H_∗ (Y; \mathbbF) → H_∗ (X; \mathbbF) (induced by the inclusion map) is injective. This notion was introduced by Kühnel in [10]. In this paper we prove the following two combinatorial criteria for tightness. (a) Any (k+1)-neighbourly k-stacked \mathbbF-homology manifold with boundary is \mathbbF-tight. Also, (b) any \mathbbF-orientable (k+1)-neighbourly k-stacked \mathbbF-homology manifold without boundary is \mathbbF-tight, at least if its dimension is not equal to 2k+1. The result (a) appears to be the first criterion to be found for tightness of (homology) manifolds with boundary. Since every (k+1)-neighbourly k-stacked manifold without boundary is, by definition, the boundary of a (k+1)-neighbourly k-stacked manifold with boundary - and since we now know several examples (including two infinite families) of triangulations from the former class - theorem (a) provides us with many examples of tight triangulated manifolds with boundary. The second result (b) generalizes a similar result from [2] which was proved for a class of combinatorial manifolds without boundary. We believe that theorem (b) is valid for dimension 2k+1 as well. Except for this lacuna, this result answers a recent question of Effenberger [8] affirmatively.