A dimension-independent simplicial data structure for non-manifold shapes

We consider the problem of representing and manipulating non-manifold multi-dimensional shapes, discretized as $d$-dimensional simplicial Euclidean complexes, for modeling finite element meshes derived from CAD models. We propose a dimension-independent data structure for simplicial complexes, that we call the {\em Incidence Simplicial (IS)} data structure. The IS data structure is scalable to manifold complexes, and supports efficient traversal and update algorithms for performing topological modifications, such as hole removal or dimension reduction. It has the same expressive power and performances as the incidence graph, commonly used for dimension-independent representation of simplicial and cell complexes, but it is much more compact. We present efficient algorithms for traversing, generating and updating a simplicial complex described as an IS data structure. We compare the IS data structure with dimension-independent and dimension-specific representations for simplicial complexes. Finally, we briefly discuss two applications that the IS data structure supports, namely decomposition of non-manifold objects for effective geometric reasoning, and multi-resolution modeling of non-manifold multi-dimensional shapes

Hodge filtered complex bordism

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and \A^1-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.Comment: minor revision; final version accepted for publication by the Journal of Topolog

Moment-angle complexes, monomial ideals, and Massey products

Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio

Algebraic Topology

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook \emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael Grinfeld from the University of Strathclyd

Branched Coverings, Triangulations, and 3-Manifolds

A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem of Hilden and Montesinos. The branched coverings introduced admit a rich theory in which the group of projectivities plays a central role.Comment: v2: several changes to the text body; minor correction