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 ↔ Topological ↔ Combinatorial Let G be a group and A a G -module. For each n ≥ 0 , let C n ( G, A ) = { σ | σ : G n → A } and define ∂ n : C n ( G, A ) → C n +1 ( G, A ) by ∂ n ( σ )( g 1 , . . . , g n +1 ) = g 1 σ ( g 2 , . . . , g n +1 ) n � ( − 1) i σ ( g 1 , . . . , g i g i +1 , . . . , g n +1 ) + i =1 ( − 1) n +1 σ ( g 1 , . . . , g n ) . + Since ∂ n +1 ∂ n = 0 , we obtain a cochain complex. The n th cohomology of G with coefficients in A is defined as H n ( G, A ) = Ker( ∂ n ) / Im( ∂ n − 1 ) . Mahender Singh IISER Mohali
Well-known results Cohomology of groups have concrete group theoretic interpretations. H 0 ( G, A ) = A G . H 1 ( G, A ) = Derivations/Principal 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 H 2 ( G, A ) and E ( G, A ) . There are also group theoretic interpretations of the functors H n for n ≥ 3 . Mahender Singh IISER Mohali
Symmetric extensions Let Φ : H 2 ( G, A ) → E ( G, A ) be the one-one correspondence. Under Φ , the trivial element of H 2 ( 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 H 2 ( G, A ) corresponds to S ( G, A ) under Φ ? Mahender Singh IISER Mohali
Examples of symmetric extensions Consider the non-split extension i π E 1 : 0 → Z → Z × Z / 2 → Z / 4 → 0 , where i ( n ) = (2 n, n ) and π ( n, m ) = n + 2 m . 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 [ E 1 ] ∈ S ( Z / 4 , Z ) . Consider the split extension i π E 2 : 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 [ E 2 ] ∈ S ( Z / 4 , Z ) . Mahender Singh IISER Mohali
Examples of non-symmetric extensions Consider the non-split extensions i π E 3 : 0 → Z → Z → Z / 4 → 0 , where i ( n ) = 4 n and π ( n ) = n and i ′ π E 4 : 0 → Z → Z → Z / 4 → 0 , where i ′ ( n ) = − 4 n and π ′ ( n ) = n . These extensions do not admit any symmetric section and hence [ E 3 ] , [ E 4 ] ∈ E ( Z / 4 , Z ) − S ( Z / 4 , Z ) . Thus S ( G, A ) � = 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
Action of Σ n +1 on C n ( G, A ) For n ≥ 0 , let Σ n +1 be the symmetric group on n + 1 symbols. For 1 ≤ i ≤ n , let τ i = ( i, i + 1) . For σ ∈ C n ( G, A ) and ( g 1 , . . . , g n ) ∈ G n , define g − 1 � � ( τ 1 σ )( g 1 , . . . , g n ) = − g 1 σ 1 , g 1 g 2 , g 3 , . . . , g n , g 1 , . . . , g i − 2 , g i − 1 g i , g − 1 � � ( τ i σ )( g 1 , . . . , g n ) = − σ , g i g i +1 , g i +2 , . . . , g n i for 1 < i < n , g 1 , g 2 , g 3 , . . . , g n − 1 g n , g − 1 � � ( τ n σ )( g 1 , . . . , g n ) = − σ . n It is easy to see that � � τ i τ i ( σ ) = σ, � � � � τ i τ j ( σ ) = τ j τ i ( σ ) for j � = i ± 1 , � � �� � � �� τ i τ i +1 τ i ( σ ) = τ i +1 τ i τ i +1 ( σ ) . Thus there is an action of Σ n +1 on C n ( G, A ) . Mahender Singh IISER Mohali
Compatibility of action Define d j : C n ( G, A ) → C n +1 ( G, A ) by d 0 ( σ )( g 1 , . . . , g n +1 ) = g 1 σ ( g 2 , . . . , g n +1 ) , d j ( σ )( g 1 , . . . , g n +1 ) = σ ( g 1 , . . . , g j g j +1 , . . . , g n +1 ) for 1 ≤ j ≤ n, d n +1 ( σ )( g 1 , . . . , g n +1 ) = σ ( g 1 , . . . , g n ) . Then ∂ n ( σ ) = � n +1 j =0 ( − 1) j d j ( σ ) . It turns out that τ i d j = d j τ i if i < j, τ i d j = d j τ i − 1 if j + 2 ≤ i, τ i d i − 1 = − d i , τ i d i = − d i − 1 . Let CS n ( G, A ) = C n ( G, A ) Σ n +1 . If σ ∈ CS n ( G, A ) , then it follows from the above identities that ∂ n ( σ ) ∈ CS n +1 ( G, A ) . Thus the action is compatible with coboundary operators. Mahender Singh IISER Mohali
Symmetric cohomology We obtain a cochain complex { CS n ( G, A ) , ∂ n } n ≥ 0 . Its cohomology, denoted HS n ( G, A ) , is called the symmetric cohomology of G with coefficients in A . HS 0 ( G, A ) = A G = H 0 ( G, A ) . A 1-cochain λ : G → A is symmetric if λ ( g ) = − gλ ( g − 1 ) . ZS 1 ( G, A ) = group of symmetric derivations. A 2-cochain σ : G × G → A is symmetric if σ ( g, h ) = − gσ ( g − 1 , gh ) = σ ( gh, h − 1 ) . Mahender Singh IISER Mohali
Symmetric cohomology The inclusion CS ∗ ( G, A ) ֒ → C ∗ ( G, A ) induces a homomorphism h ∗ : HS ∗ ( G, A ) → H ∗ ( G, A ) . Clearly h ∗ is an isomorphism in dimension 0 and is injective in dimension 1. Similar result holds in dimension 2. Proposition (M. Staic, J. Algebra 2009) The map h ∗ : HS 2 ( G, A ) → H 2 ( G, A ) is injective. We now have an answer to Question 1. Theorem (M. Staic, J. Algebra 2009) The map Φ ◦ h ∗ : HS 2 ( G, A ) → S ( G, A ) is a bijection. Examples in previous slides corresponds to the fact that H 2 ( Z / 4 , Z ) = Z / 4 and HS 2 ( Z / 4 , Z ) = Z / 2 . Mahender Singh IISER Mohali
Cohomology of topological groups When the group under consideration is equipped with a topology (or any other structure), it is natural to look for a cohomology theory which also takes the topology (the other structure) into account. This lead to various cohomology theories of topological groups. Topology was first inserted in the formal definition of cohomology of topological groups in the works of S. -T. Hu (1952), W. T. van Est (1953) and A. Heller (1973). Mahender Singh IISER Mohali
Continuous cohomology of topological groups Let G be a topological group and A an abelian topological group. We say that A is a topological G -module if there is a continuous action of G on A . For each n ≥ 0 , let G n be the product topological group and c ( G, A ) = { σ | σ : G n → A is a continuous map } . C n Let ∂ n : C n c ( G, A ) → C n +1 ( G, A ) be the standard coboundary map c as used for abstract groups. Then { C n c ( G, A ) , ∂ n } n ≥ 0 is a cochain complex. The cohomology of this cochain complex, denoted H ∗ c ( G, A ) , is called the continuous cohomology of G with coefficients in A . Clearly this cohomology coincides with the ordinary cohomology when the groups under consideration are discrete (in particular finite). The low dimensional cohomology groups are as expected with c ( G, A ) = A G and Z 1 H 0 c ( G, A ) = group of continuous derivations . Mahender Singh IISER Mohali
� � Extensions of topological groups An extension of topological groups i π 0 → A → E → G → 1 is an algebraically exact sequence of topological groups with the additional property that i is closed continuous and π is open continuous. If we assume that i and π are only continuous, then A viewed as a subgroup of E may not have the relative topology and the isomorphism E/i ( A ) ∼ = G may not be a homeomorphism. Since i is closed continuous, its an embedding of A onto a closed subgroup of E . → E ′ π ′ i ′ i π Two extensions 0 → A → E → G → 1 and 0 → A → G → 1 are called equivalent if there exists an open continuous isomorphism φ : E → E ′ making the following diagram commute. i π � A � E � 1 0 G φ i ′ π ′ � A � E ′ � G � 1 0 Mahender Singh IISER Mohali
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