We compute the group cohomology H∗(G;ZG) or the reduced L-cohomology, H∗(G;L2G) when G is either • a graph product of infinite groups, • an Artin group, or • a Bestvina-Brady group. The proofs will appear in [8]. Suppose (W,S) is a Coxeter system. A subset T ⊂ S is spherical if it generates a finite subgroup of W . Let S denote the poset of spherical subsets of S. The nerve of (W,S) is the simp...

In this paper, we analyze the structure possessed by the homotopy groups of a simplicial abelian Hopf algebra over the field F2. Specifically, we review the higher-order structure that the homotopy groups of a simplicial commutative algebra and simplicial cocommutative coalgebra possess. We then demonstrate how these structures interact under the added conditions present in a Hopf algebra. Intr...

This is a course in algebraic topology for anyone who has seen the fundamental group and homology. Traditionally (20 years ago), the syllabus included cohomology, the universal coefficient theorem, products, Tor and Ext and duality. We will cover all those topics, while also building a modern framework of derived categories and derived functors using the Cartan–Eilenberg method. We will apply t...

The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work wit...

In this paper we explore the relationships between the motivic and simplicial cohomology operations defined on mod-l motivic cohomology. We also explore similar relationships in étale cohomology and conclude by considering certain operations that commute with proper push-forwards.

We show that the de Rham theorem, interpreted as the isomorphism between distributional de Rham cohomology and simplicial homology in the dual dimension for a simplicial decomposition of a compact oriented manifold, is a straightforward consequence of elementary properties of currents. The explicit construction of this isomorphism extends to other cases, such as relative and absolute cohomology...

Simplicial volume is a homotopy invariant of oriented closed connected manifolds introduced by Gromov in his proof of Mostow rigidity. On the one hand, the simplicial volume of a Riemannian manifold encodes non-trivial information about the Riemannian volume; on the other hand, simplicial volume can be described in terms of a certain functional analytic version of homological algebra (bounded c...

In this paper we introduce and study a cohomology theory {Hn(−,A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A, n)}n≥0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−, n) is right adjoint to the functor ℘n, w...

the aim of this paper is to study the $(alpha‎, ‎gamma)$-prolongation of central extensions‎. ‎we obtain the obstruction theory for $(alpha‎, ‎gamma)$-prolongations and classify $(alpha‎, ‎gamma)$-prolongations thanks to low-dimensional cohomology groups of groups‎.