Enumerative properties of triangulations of spherical bundles over S^1

We give a complete characterization of all possible pairs (v,e), where v is the number of vertices and e is the number of edges, of any simplicial triangulation of an S^k-bundle over S^1. The main point is that Kuhnel's triangulations of S^{2k+1} x S^1 and the nonorientable S^{2k}-bundle over S^1 are unique among all triangulations of (n-1)-dimensional homology manifolds with first Betti number nonzero, vanishing second Betti number, and 2n+1 vertices.Comment: To appear in European J. of Combinatorics. Many typos fixe

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

Minimal triangulations of sphere bundles over the circle

For integers $d \geq 2$ and $\epsilon = 0$ or 1, let $S^{1, d - 1}(\epsilon)$ denote the sphere product $S^{1} \times S^{d - 1}$ if $\epsilon = 0$ and the twisted $S^{d - 1}$ bundle over $S^{1}$ if $\epsilon = 1$. The main results of this paper are: (a) if $d \equiv \epsilon$ (mod 2) then $S^{1, d - 1}(\epsilon)$ has a unique minimal triangulation using $2d + 3$ vertices, and (b) if $d \equiv 1 - \epsilon$ (mod 2) then $S^{1, d - 1}(\epsilon)$ has minimal triangulations (not unique) using $2d + 4$ vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while $S^{1, d - 1}(\epsilon)$ has at most one $(2d + 3)$-vertex triangulation (one if $d \equiv \epsilon$ (mod 2), zero otherwise), in sharp contrast, the number of non-isomorphic $(2d + 4)$-vertex triangulations of these $d$-manifolds grows exponentially with $d$ for either choice of $\epsilon$. The result in (a), as well as the minimality part in (b), is a consequence of the following result: (c) for $d \geq 3$, there is a unique $(2d + 3)$-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension $d$. This amazing simplicial complex was first constructed by K\"{u}hnel in 1986. Generalizing a 1987 result of Brehm and K\"{u}hnel, we prove that (d) any triangulation of a non-simply connected closed $d$-manifold requires at least $2d + 3$ vertices. The result (c) completely describes the case of equality in (d). The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality.Comment: 15 pages, Revised, To appear in `Journal of Combinatorial Theory, Ser. A

Minimal flag triangulations of lower-dimensional manifolds

We prove the following results on flag triangulations of 2- and 3-manifolds. In dimension 2, we prove that the vertex-minimal flag triangulations of $\mathbb{R} P^2$ and $\mathbb{S}^1\times \mathbb{S}^1$ have 11 and 12 vertices, respectively. In general, we show that $8+3k$ (resp. $8+4k$) vertices suffice to obtain a flag triangulation of the connected sum of $k$ copies of $\mathbb{R} P^2$ (resp. $\mathbb{S}^1\times \mathbb{S}^1$). In dimension 3, we describe an algorithm based on the Lutz-Nevo theorem which provides supporting computational evidence for the following generalization of the Charney-Davis conjecture: for any flag 3-manifold, $\gamma_2:=f_1-5f_0+16\geq 16 \beta_1$, where $f_i$ is the number of $i$-dimensional faces and $\beta_1$ is the first Betti number over a field. The conjecture is tight in the sense that for any value of $\beta_1$, there exists a flag 3-manifold for which the equality holds.Comment: 6 figures, 3 tables, 19 pages. Final version with a few typos correcte

Convex and Algebraic Geometry

The subjects of convex and algebraic geometry meet primarily in the theory of toric varieties. Toric geometry is the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Recent developments in toric geometry that were discussed during the workshop include applications to mirror symmetry, motivic integration and hypergeometric systems of PDE’s, as well as deformations of (unions of) toric varieties and relations to tropical geometry

Lagrangian Floer potential of orbifold spheres

For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov–Witten potential, which serves as the quantum-corrected Landau–Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a consequence we obtain an enumerative meaning of mirror maps for elliptic curve quotients. Furthermore, we prove that the open Gromov–Witten potential is convergent, even in the general-type cases, and has an isolated singularity at the origin, which is an important ingredient of proving homological mirror symmetry.National Research Foundation of Korea; 2010-0019516; 2012R1A1A2003117; 2013R1A1A1058646 - National Research Foundation of Kore