Deformations of bordered Riemann surfaces and associahedral polytopes

We consider the moduli space of bordered Riemann surfaces with boundary and marked points. Such spaces appear in open-closed string theory, particularly with respect to holomorphic curves with Lagrangian submanifolds. We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. Our main result classifies all such spaces that can be realized as convex polytopes. A new polytope is introduced based on truncations of cubes, and its combinatorial and algebraic structures are related to generalizations of associahedra and multiplihedra.Comment: 25 pages, 31 figure

Remarks on the combinatorial intersection cohomology of fans

We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric $g$ and $h$ polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two complex gives the rigidity complex previously used by Kalai to study $g_2$. We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that $g_k(P) = 0$ implies $g_k(P^*) = 0$ and $g_{k+1}(P) = 0$.Comment: 34 pages. Typos fixed; final version, to appear in Pure and Applied Math Quarterl

LSLS-Category of Moment-Angle Manifolds, Massey Products, and a Generalization of the Golod Property

We give various bounds for the Lusternik-Schnirelmann category of moment-angle complexes and show how this relates to vanishing of Massey products in $\mathrm{Tor}^+_{R[v_1,\ldots,v_n]}(R[K],R)$. In particular, we characterise the Lusternik-Schnirelmann category of moment-angle manifolds $\mathcal{Z}_K$ over triangulated $d$-spheres $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. This characterisation is given in terms of the combinatorics of $K$, the cup product length of $H^*(\mathcal{Z}_K)$, as well as a certain generalisation of the Golod property. Some applications include information about the category and vanishing of Massey products for moment-angle complexes over fullerenes and $k$-neighbourly complexes.Comment: New examples adde

PALP - a User Manual

This article provides a complete user's guide to version 2.1 of the toric geometry package PALP by Maximilian Kreuzer and others. In particular, previously undocumented applications such as the program nef.x are discussed in detail. New features of PALP 2.1 include an extension of the program mori.x which can now compute Mori cones and intersection rings of arbitrary dimension and can also take specific triangulations of reflexive polytopes as input. Furthermore, the program nef.x is enhanced by an option that allows the user to enter reflexive Gorenstein cones as input. The present documentation is complemented by a Wiki which is available online.Comment: 71 pages, to appear in "Strings, Gauge Fields, and the Geometry Behind - The Legacy of Maximilian Kreuzer". PALP Wiki available at http://palp.itp.tuwien.ac.at/wiki/index.php/Main_Pag

Geometric and Topological Combinatorics

The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions

Algebraic Topology of Calabi-Yau Threefolds in Toric Varieties

We compute the integral homology (including torsion), the topological K-theory, and the Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and complete intersections in Gorenstein toric Fano varieties. The methods are purely topological