This paper investigates properties of finite sheeted covering spaces of arithmetic hyperbolic 3-orbifolds (see §2). The main motivation is a central unresolved question in the theory of closed hyperbolic 3-manifolds; namely whether a closed hyperbolic 3-manifold is virtually Haken. Various strengthenings of this have also been widely studied. Of specific to interest to us is the question of whe...

Abstract Topology is the foundation for many industrial applications ranging from CAD to simulation analysis. Computational topology mostly focuses on structured data such as mesh. However, unstructured datasets point sets remain a virgin land scientists. The significance of point-based can never be overemphasized, especially in area reverse engineering, geometric modeling, and algorithmic In t...

In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vrećica, and Z̆ival...

We give a complete characterization of all possible pairs (f0, f1), where f0 is the number of vertices and f1 is the number of edges, of any simplicial triangulation of an Sk-bundle over S1. The main point is that Kühnel’s triangulations of S2k+1 × S1 and the nonorientable S2k-bundle over S1 are unique among all triangulations of (n − 1)-dimensional homology manifolds with first Betti number no...

In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one. We also give algorithms ...