From 3-algebras to Δ-groups and symmetric cohomology

Abstract

We introduce -groups and show how they fit in the context of lattice field theory. To a topological space M we associate a -group Γ (M). We define the symmetric cohomology HS n (G, A) of a group G with coefficients in a G-module A. The -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)).

Symmetric cohomology of groups in low dimension

“…In [4] it was constructed an action of Σ n+1 on C n (G, A) (for every n) and it was proved that it is compatible with the differential.…”

“…, g n−1 g n , (g n ) −1 ). [3,4]) The above formulas define an action of Σ n+1 on C n (G, A) which is compatible with the differential ∂. Definition 2.3.…”

Crossed modules and symmetric cohomology of groups

“…In [8] Staic introduced a subcomplex CS * (G, M ) ⊂ C * (G, M ), whose homology is known as the symmetric cohomology of G with coefficients in M and is denoted by HS * (G, M ). The definition is based on an action of Σ n+1 on C n (G, M ) (for every n) compatible with the differential.…”

“…Let G be a group and M be a G-module. Symmetric cohomology HS * (G, M ) was introduced by M. Staic [8] as a variant of classical group cohomology. A. Zarelua's prior definition [11] of exterior cohomology H * λ (G, M ) is very closely related to this, as shown in [6].…”

Symmetric cohomology of groups as a Mackey functor

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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.

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