Symmetric cohomology of groups
Mahender Singh
Knots, Braids and Automorphism Groups Novosibirsk July 2014
Mahender Singh IISER Mohali Introduction
Cohomology of groups is a contravariant functor turning groups and modules over groups into graded abelian groups. It came into being with the fundamental work of Eilenberg and MacLane (Ann. Math. 1947). The theory was further developed by Hopf, Eckmann, Segal, Serre, and many other mathematicians. It has been studied from different perspectives with applications in various areas of mathematics. It provides a beautiful link between algebra and topology.
Mahender Singh IISER Mohali Cohomology of groups
There are three main equivalent descriptions of cohomology of groups.
Algebraic ↔ T opological ↔ Combinatorial
Let G be a group and A a G-module. For each n ≥ 0, let Cn(G, A) = {σ | σ : Gn → A} and define ∂n : Cn(G, A) → Cn+1(G, A) by
n ∂ (σ)(g1, . . . , gn+1) = g1σ(g2, . . . , gn+1) n X i + (−1) σ(g1, . . . , gigi+1, . . . , gn+1) i=1 n+1 + (−1) σ(g1, . . . , gn).
Since ∂n+1∂n = 0, we obtain a cochain complex. The nth cohomology of G with coefficients in A is defined as Hn(G, A) = Ker(∂n)/ Im(∂n−1).
Mahender Singh IISER Mohali Well-known results
Cohomology of groups have concrete group theoretic interpretations. H0(G, A) = AG. H1(G, A) = Derivations/P rincipal Derivations. Let E(G, A) = Set of equivalence classes of extensions of G by A giving rise to the given action of G on A. Then there is a one-one correspondence between H2(G, A) and E(G, A). There are also group theoretic interpretations of the functors Hn for n ≥ 3.
Mahender Singh IISER Mohali Symmetric extensions
Let Φ: H2(G, A) → E(G, A) be the one-one correspondence. Under Φ, the trivial element of H2(G, A) corresponds to the equivalence class of an extension
0 → A → E → G → 1
admitting a section s : G → E which is a group homomorphism. An extension E : 0 → A → E → G → 1 of G by A is called a symmetric extension if there exists a section s : G → E such that s(g−1) = s(g)−1 for all g ∈ G. Such a section is called a symmetric section. Let S(G, A) = [E] ∈ E(G, A) | E is a symmetric extension . Then the following question seems natural. Question 1 What elements of H2(G, A) corresponds to S(G, A) under Φ?
Mahender Singh IISER Mohali Examples of symmetric extensions
Consider the non-split extension
i π E1 : 0 → Z → Z × Z/2 → Z/4 → 0, where i(n) = (2n, n) and π(n, m) = n + 2m. Let s : Z/4 → Z × Z/2 be given by s(0) = (0, 0), s(1) = (−1, 1), s(2) = (0, 1) and s(3) = (1, 1).
Then s is a symmetric section and hence [E1] ∈ S(Z/4, Z). Consider the split extension
i π E2 : 0 → Z → Z × Z/4 → Z/4 → 0,
where i(n) = (n, 0) and π(n, m) = m. Then s : Z/4 → Z × Z/4 given by s(m) = (0, m) is a symmetric section and hence [E2] ∈ S(Z/4, Z).
Mahender Singh IISER Mohali Examples of non-symmetric extensions
Consider the non-split extensions
i π E3 : 0 → Z → Z → Z/4 → 0, where i(n) = 4n and π(n) = n and
i0 π E4 : 0 → Z → Z → Z/4 → 0, where i0(n) = −4n and π0(n) = n. These extensions do not admit any symmetric section and hence [E3], [E4] ∈ E(Z/4, Z) − S(Z/4, Z). Thus S(G, A) 6= E(G, A) in general.
Mahender Singh IISER Mohali Symmetric cohomology
Mihai Staic (2009) answered Question 1 for abstract groups. Motivated by some problems in constructing invariants of 3-manifolds, Staic introduced a new cohomology theory of groups called symmetric cohomology which classifies symmetric extensions in dimension two.
Mahender Singh IISER Mohali n Action of Σn+1 on C (G, A)
For n ≥ 0, let Σn+1 be the symmetric group on n + 1 symbols.
For 1 ≤ i ≤ n, let τi = (i, i + 1). n n For σ ∈ C (G, A) and (g1, . . . , gn) ∈ G , define