We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse $X\searrow Y$ of finite spaces induces a simplicial collapse $\k(X)\searrow \k(Y)$ of their associated simplicial c...

We introduce notions of combinatorial blowups, building sets, and nested sets for arbitrary meet-semilattices. This gives a common abstract framework for the incidence combinatorics occurring in the context of De Concini-Procesi models of subspace arrangements and resolutions of singularities in toric varieties. Our main theorem states that a sequence of combinatorial blowups, prescribed by a b...

We prove a limit theorem for extension theory for metric spaces. This theorem can be put in the following way. Suppose that K is a simplicial complex, jKj is given the weak topology, and a metrizable space X is the limit of an inverse sequence of metrizable spaces X i having the property that X i jKj for each i 2 N. Then XjKj. This latter property means that for each closed subset A of X and ma...

Using a simplex-crossing counting technique we prove: if the number of non-improperly intersecting simplices with vertices in a set S of n labeled points in ~d is O(nra/2]) , then there are 2 °('~rd/21) different geometric simplicial complexes with vertices in S. © 1997 Elsevier Science B.V.

We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the no...

This paper starts with an observation that two infinite series of simplicial complexes, which a priori do not seem to have anything to do with each other, have the same homotopy type. One series consists of the complexes of directed forests on a double directed string, while the other one consists of Shapiro–Welker models for the spaces of hyperbolic polynomials with a triple root. We explain t...

We show how the flag f-vector of a polytope changes when cutting off any face, generalizing work of Lee for simple polytopes. The result is in terms of explicit linear operators on cd-polynomials. Also, we obtain the change in the flag f-vector when contracting any face of the polytope.

Let ∆n−1 denote the (n − 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of ∆n−1 obtained by starting with the full (d − 1)-dimensional skeleton of ∆n−1 and then adding each d-simplex independently with probability p = c n . We compute an explicit constant γd, with γ2 ' 2.45, γ3 ' 3.5 and γd = Θ(log d) as d → ∞, so that for c < γd such a random simplicial complex either colla...

We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition. These ideas are central to constructing realization...

simpcomp is an extension to GAP, the well known system for computational discrete algebra. It allows the user to work with simplicial complexes. In the latest version, support for simplicial blowups and discrete normal surfaces was added, both features unique to simpcomp. Furthermore, new functions for constructing certain infinite series of triangulations have been implemented and interfaces t...