The Joyal model structure on simplicial sets is extended to a model structure on the simplicial presheaves on a small site, in which the cofibrations are monomorphisms and the weak equivalences are local (or stalkwise) Joyal equivalences. The model structure is shown to be left proper.

A simplicial complex X is d-Leray if H̃i(Y ;Q) = 0 for all induced subcomplexes Y ⊂ X and i ≥ d. Let L(X) denote the minimal d such that X is d-Leray. Theorem: Let X,Y be simplicial complexes on the same vertex set. Then L(X ∩ Y ) ≤ L(X) + L(Y ) , L(X ∪ Y ) ≤ L(X) + L(Y ) + 1 .

The construction of the Bier sphere Bier(K) for a simplicial complex K is due to Bier (1992). Björner, Paffenholz, Sjöstrand and Ziegler (2005) generalize this construction to obtain a Bier poset Bier(P, I) from any bounded poset P and any proper ideal I ⊆ P . They show shellability of Bier(P, I) for the case P = Bn, the boolean lattice, and thereby obtain ‘many shellable spheres’ in the sense ...

Definition 20 (simplex). A k-simplex σ is the convex hull of a set P of k + 1 affinely independent points. In particular, a 0-simplex is a vertex, a 1-simplex is an edge, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron. A k-simplex is said to have dimension k. A face of σ is a simplex that is the convex hull of a nonempty subset of P. Faces of σ come in all dimensions from zero (σ’s...

In this thesis we discuss some new results concerning the combinatorial Laplace operator of a simplicial complex. The combinatorial Laplacian of a simplicial complex encodes information about the relationships between adjacent simplices in the complex. This thesis is divided into two relatively disjoint parts. In the first portion of the thesis, we derive a relationship between the Laplacian sp...

Let G be a finite group, p be a prime, k be an algebraically closed field of characteristic p. J. L. Alperin has conjectured that for any finite group G, # (isomorphism types of simple fcG-modules) = £ # (isomorphism types of projective simple A: (C) where the sum is taken over a set of representatives of the conjugacy classes of /7-subgroups of G. There is also a ' blockwise' version of this c...

This paper characterizes a class of boolean functions, designated of hard-threshold type, in terms of their action on the underlying space graph.

We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.

In this paper, we characterize all possible h-vectors of 2-dimensional Buchsbaum simplicial complexes.

We reprove and generalize in a combinatorial way the result of A. Björner, [Bj81, Theorem 3.3], that order complexes of noncomplemented lattices are contractible, namely by showing that these simplicial complexes are in fact nonevasive, in particular collapsible.