Lattice gauge theory with gauge group ZP is reconsidered in four dimensions on a simplicial complex K. One finds that the dual theory, formulated on the dual block complex K̂, contains topological modes which are in correspondence with the cohomology group H2(K̂, ZP ), in addition to the usual dynamical link variables. This is a general phenomenon in all models with single plaquette based actions...

Given a Coxeter system .W;S/ and a positive real multiparameter q , we study the “weighted L–cohomology groups,” of a certain simplicial complex † associated to .W;S/ . These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to .W;S/ and the multiparameter q . They have a “von Neumann dimension” with respect to the associated “Hecke–von Neumann algebra” ...

We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These generators are constructed using the de Rham representatives for the cohomology of classifying spaces BK where K is a compact Lie group; such representatives (un...

In this article we give characteristic properties of the simplicial homology groups of digital images which are based on the simplicial homology groups of topological spaces in algebraic topology. We then investigate EilenbergSteenrod axioms for the simplicial homology groups of digital images. We state universal coefficient theorem for digital images. We conclude that the Künneth formula for s...

Generalizing the notion of a Koszul algebra, a graded kalgebra A is K2 if its Yoneda algebra ExtA(k, k) is generated as an algebra in cohomology degrees 1 and 2. We prove a strong theorem about K2 factor algebras of Koszul algebras and use that theorem to show the Stanley-Reisner face ring of a simplicial complex ∆ is K2 whenever the Alexander dual simplicial complex ∆∗ is (sequentially) Cohen-...

Associated to every finite simplicial complex K there is a “moment-angle” finite CW-complex, ZK ; if K is a triangulation of a sphere, ZK 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 combinatori...

While the Euler characteristic χ(G) = ω1(G) = ∑ x ω(x) super counts simplices, the Wu characteristics ωk(G) = ∑ x1∼x2···∼xk ω(x1) · · ·ω(xk) super counts simultaneously pairwise interacting k-tuples of simplices in a finite abstract simplicial complex G. More generally, one can define the k-intersection number ωk(G1, . . . Gk), which is the same sum but where xi ∈ Gi. For every k ≥ 1 we define ...

Before starting, we’ll say a few words of motivation. We wish to understand the infinite Galois groupG = Gal(Q/Q), which is very important in number theory. In particular, we wish to understand G-modules, namely modules over some ring that are endowed with an action of G. Such objects tend to be complicated. Recall that in algebraic topology, we define the homology and cohomology of simplicial ...

The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel’s conjecture ...

Triangulation is a convenient tool to consider when we are determining some properties of a topological space. Specifically, we can compute the homology and cohomology groups using the simplicial homology and cohomology of the triangulated space which is much easier than involving more complicated theories. In this paper, we will review the standard proof by Radó of the existence of triangulati...