Stacked polytopes and tight triangulations of manifolds

Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise linear embedding of the triangulation into euclidean space is "as convex as possible". It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be a purely combinatorial condition which implies the tightness of a triangulation. Here we present other sufficient and purely combinatorial conditions which can be applied to the odd-dimensional case as well. One of the conditions is that all vertex links are stacked spheres, which implies that the triangulation is in Walkup's class $\mathcal{K}(d)$. We show that in any dimension $d\geq 4$ \emph{tight-neighborly} triangulations as defined by Lutz, Sulanke and Swartz are tight. Furthermore, triangulations with $k$-stacked vertex links and the centrally symmetric case are discussed.Comment: 28 pages, 2 figure

Hamiltonian submanifolds of regular polytopes

We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called {\it super-neighborly triangulations}) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the $d$-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of $S^2 \times S^2$. By this example all regular cases of $n$ vertices with $n < 20$ or, equivalently, all cases of regular $d$-polytopes with $d\leq 9$ are now decided.Comment: 26 pages, 4 figure

On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of ${\cal W}_k(d)$. We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of its Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of ${\cal W}_k(d)$ for $d\geq 2k$. As one consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, as well as determine the integral homology type of members of ${\cal W}^{\ast}_k(d)$ for $d \geq 2k+2$. As another application, we prove that, when $d \neq 2k+1$, all members of ${\cal W}^{\ast}_k(d)$ are tight. We also characterize the tight members of ${\cal W}^{\ast}_k(2k + 1)$ in terms of their $k^{\rm th}$ Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for triangulated manifolds in which the members of ${\cal W}_1(d)$ provide the equality case. This generalises a result (the $d=4$ case) due to Walkup and Kuehnel. As a consequence, it is shown that every tight member of ${\cal W}_1(d)$ is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuehnel and Lutz asserting that tight triangulated manifolds should be strongly minimal.Comment: Revised version, 22 pages. arXiv admin note: substantial text overlap with arXiv:1102.085

Invariants in Low-Dimensional Topology and Knot Theory

This meeting concentrated on topological invariants in low dimensional topology and knot theory. We include both three and four dimensional manifolds in our phrase “low dimensional topology”. The intent of the conference was to understand the reach of knot theoretic invariants into four dimensions, including results in Khovanov homology, variants of Floer homology and quandle cohomology and to understand relationships among categorification, topological quantum field theories and four dimensional manifold invariants as in particular Seiberg-Witten invariants

Classification of tight contact structures on small Seifert fibered L-spaces

The Ozsvath-Szabo contact invariant is a complete classification invariant for tight contact structures on small Seifert fibered 3-manifolds which are L-spaces.Comment: 30 pages, 3 figure

Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangulations include the Brieskorn homology spheres $\Sigma (p,q,r)$, the lens spaces $\operatorname{L} (q,1)$ and, as a limit case, $(\mathbf{S}^2 \times \mathbf{S}^1)^{\# (p-1)(q-1)}$.Comment: 28 pages, 9 figures. Minor update. To appear in Israel Journal of Mathematic