Symmetric Cohomology Research Articles

Symmetric cohomology and symmetric Hochschild cohomology of cocommutative Hopf algebras

Staic defined symmetric cohomology of groups and studied that the secondary symmetric cohomology group is corresponding to group extensions and the injectivity of the canonical map from symmetric cohomology to classical cohomology. In this paper, we define symmetric cohomology and symmetric Hochschild cohomology for cocommutative Hopf algebras. The first one is a generalization of symmetric cohomology of groups. We give an isomorphism between symmetric cohomology and symmetric Hochschild cohomology, which is a symmetric version of the classical result about cohomology of groups by Eilenberg and MacLane and cohomology of Hopf algebras by Ginzburg and Kumar. Moreover, to consider the condition that symmetric cohomology coincides with classical cohomology, we investigate the projectivity of a resolution which gives symmetric cohomology.

Symmetric cohomology of groups and Poincaré duality

We prove a Poincaré duality theorem for finite groups, where the (co)homologies involved are a variant of classical group cohomology due to Zarelua and Staic, denoted by Hλ⁎. In particular, we show that in certain cases (but not always), for a finite group G of order n and a G-module M, we have the isomorphismHλn−i−1(G,M)→≅Hiλ(G,Mtw), where Mtw is a twisting of M defined in Section 5.

Symmetric Hochschild cohomology of twisted group algebras

We show that there is an action of the symmetric group on the Hochschild cochain complex of a twisted group algebra with coefficients in a bimodule. This allows us to define the symmetric Hochschild cohomology of twisted group algebras, similarly to th construction of symmetric group cohomology due to Staic. We give explicit embeddings and connecting homomorphisms between the symmetric cohomology spaces and symmetric Hochschild cohomology of twisted group algebras.

Four-cocycles in commutative semigroup cohomology

The symmetric cohomology group of a commutative semigroup is defined in dimension 4 and is shown to equal its triple cohomology group.

Symmetric cohomology of groups

We investigate the relationship between the symmetric, exterior and classical cohomologies of groups. The first two theories were introduced respectively by Staic and Zarelua. We show in particular, that there is a map from exterior cohomology to symmetric cohomology which is a split monomorphism in general and an isomorphism in many cases, but not always. We introduce two spectral sequences which help to explain the relationship between these cohomology groups. As a sample application we obtain that symmetric and classical cohomologies are isomorphic for torsion free groups.

Symmetric cohomology of groups as a Mackey functor

Symmetric cohomology of groups, defined by M. Staic in [2], is similar to the way one defines the cyclic cohomology for algebras. We show that there is a well-defined restriction, conjugation and transfer map in symmetric cohomology, which form a Mackey functor under a restriction. Some new properties for the symmetric cohomology group using normalized cochains are also given.

Symmetric continuous cohomology of topological groups

In this paper, we introduce a symmetric continuous cohomology of topological groups. This is obtained by topologizing a recent construction due to Staic (J. Algebra 322 (2009), 1360-1378), where a symmetric cohomology of abstract groups is constructed. We give a characterization of topological group extensions that correspond to elements of the second symmetric continuous cohomology. We also show that the symmetric continuous cohomology of a profinite group with coefficients in a discrete module is equal to the direct limit of the symmetric cohomology of finite groups. In the end, we also define symmetric smooth cohomology of Lie groups and prove similar results.

Symmetric cohomology of groups in low dimension

We give an explicit characterization for group extensions that correspond to elements of the symmetric cohomology HS2(G, A). We also give conditions for the map HSn(G, A) → Hn(G, A) to be injective.

From 3-algebras to Δ-groups and symmetric cohomology

We introduce Δ-groups and show how they fit in the context of lattice field theory. To a topological space M we associate a Δ-group Γ ( M ) . We define the symmetric cohomology HS n ( G , A ) of a group G with coefficients in a G-module A. The Δ-group Γ ( M ) is determined by the action of π 1 ( M ) on π 2 ( M ) and an element of HS 3 ( π 1 ( M ) , π 2 ( M ) ) .

Homotopy classification of graded Picard categories

Certain low-dimensional symmetric cohomology groups of G-modules, for any given group G, are computed as the cohomology of an explicit cochain complex. This result is used to establish natural one-to-one correspondences between elements of the 3rd symmetric cohomology groups of G-modules, G-equivariant pointed 2-connected homotopy 4-types, and equivalence classes of G-graded Picard categories. The simplicial nerve of a G-graded Picard category is also constructed and studied.

Projective limits of locally symmetric spaces and cohomology.

Projective limits of locally symmetric spaces and cohomology.

Crossed modules and symmetric cohomology of groups

This paper links the third symmetric cohomology (introduced by Staic and Zarelua ) to crossed modules with certain properties. The equivalent result in the language of 2-groups states that an extension of 2-groups corresponds to an element of $HS^3$ iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding $HS^2$ and abelian extensions of groups.