For a simplicial complex $K$ with $m$ vertices, there is canonical $\mathbb Z_2^m$-space known as real moment angle R \mathcal Z_K$. In this paper, we consider the quotient spaces $Y=\mathbb Z_K / \mathbb Z_2^{k}$, where pure shellable and Z_2^k \subset Z_2^m$ maximal free action on A typical example of such small cover, cover topological analog toric manifold. We compute integral cohomology gr...

We prove a Banach version of Garland's method proving vanishing cohomology for groups acting on simplicial complexes. The novelty this new is that our condition applies to every reflexive space. This allows us deduce several criteria group with coefficients in classes spaces (uniformly curved spaces, Hilbertian and $L^p$ spaces). Using these criteria, we improve recent results regarding fixed p...

The purpose of this paper is to give a characterization families expander graphs via right-angled Artin groups. We prove that sequence simplicial [Formula: see text] forms family if and only certain natural mini-max invariant arising from the cup product in cohomology rings groups agrees with Cheeger constant graphs, thus allowing us characterize cohomology. This result proved more general fram...

Background and Motivation 1 1. Sheaves and Stacks 3 1.1. Sheaves 3 1.2. Stacks 4 2. Sheaves on Stacks 5 2.1. The Site of a Stack 5 2.2. The site (Mell)ét 6 2.3. Homotopy Limits and Sheaves of Spectra 7 2.3.1. Limits and Homotopy Limits 7 2.3.2. Derived Limits and Corrected Homotopy Limits 9 2.3.3. Sheaves of Orthogonal Spectra, Symmetric Spectra, and S-Modules 13 3. The Descent Spectral Sequenc...

On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re...

We construct the motive of an algebraic stack in Nisnevich topology. For stacks which are locally quotient stacks, we give a presentation terms simplicial schemes. also show that for motivic cohomology agrees with Edidin-Graham-Totaro Chow groups integer coefficients.

In this survey, we discuss two research areas related to Massey’s higher operations. The first direction is connected with the cohomology of Lie algebras and theory representations. second main theme at intersection toric topology, homotopy polyhedral products, homology local rings, Stanley–Reisner rings simplicial complexes.