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The Cohomology Class of the Mod 4 Braid Group The Cohomology Class of the Mod 4 Braid Group Trevor Nakamura Abstract The mod 4 braid group, Zn, is defined to be the quotient of the braid group by its level 4 congruence subgroup. Kordek and Margalit proved Zn is an extension of the symmetric n 2 group by Z2 . For n ≥ 4, we construct a 2-cocycle in the group cohomology of the symmetric group with twisted coefficients classifying Zn. We show this cocycle is the mod 2 reduction of the 2-cocycle corresponding to the extension of the symmetric group by the abelianization of the pure braid group. We also construct the 2-cocycle corresponding to this second extension and show it represents an order two element in the cohomology of the symmetric group. Furthermore, we give presentations for both extensions and a normal generating set for the level 4 congruence subgroup of the braid group. 1 Introduction In 2019 Kordek and Margalit [8] showed the quotient of the braid group by the level 4 con- gruence subgroup is an extension of the symmetric group, Sn. Eilenberg and MacLane [6] proved we can classify group extensions by low dimensional group cohomology with twisted co- efficients. In particular, Zn is classified by the second cohomology of the symmetric group with n coefficients in Z2. We begin by constructing a CW-complex which is the two-skeleton of an Eilenberg-MacLane space for the symmetric group. This Eilenberg-MacLane space gives us a low dimensional approximation for a resolution of Z over ZSn-modules. Then we construct a presentation for the extension of the symmetric group by the abelianization of the pure braid group. We obtain our main results from classifying the extension of the symmetric group by the abelianization of the pure braid group and composing this 2-cocycle with the mod 2 reduction of integers. Let G be a group and A be a G-module. We represent the, possibly nontrivial, action of G on A by an automorphism θ ∶ A → A. The cohomology of G with coefficients in A is equivalent to the cohomology of an Eilenberg-MacLane space for G, referred to as a K(G, 1)-space, with a local 2 coefficients system of A determined by θ. We use Hθ (G; A) to denote the second cohomology group of G with coefficients in A twisted by θ. Fixing the action of G on A, Eilenberg and arXiv:2107.12263v1 [math.AT] 26 Jul 2021 2 MacLane proved there exists a bijection between Hθ (G, A) and the set of group extensions, up to equivalence, of G by A which give rise to θ [6]. In this paper we will only consider one action of the symmetric group and therefore often omit θ. We give a more explicit explanation of the background for group cohomology in section 2.2. Extensions of the symmetric group arise naturally while studying quotients of the braid group, Bn. The level m congruence subgroup of the braid group, Bn m , is the kernel of a map: [ ] Bn → GLn Zm (more information is given in section 2.3). Brendle and Margalit proved if n 4, ( ) ≥ then Bn 4 is the subgroup of PBn generated by squares of all elements [3]. Recent work by [ ] 1 n 2 Kordek and Margalit showed Zn = Bn Bn 4 is an extension of Sn by Z2 where Sn permutes n 2 ~ [ ] the generators of Z2 by the standard action on unordered pairs of 1,...,n . Let θ represent n { } 2 the automorphism corresponding to this action of Sn on Z2 , then our first theorem shows the n cocylce classifying Zn is determined by a cocycle which classifies an extension of Sn by Z 2 . n 2 2 Theorem 1.1. If n ≥ 4, then the cohomology class κ ∈ H Sn; Z2 is the mod 2 reduction n 2 [ ] ( ) of an element φ ∈ H Sn; Z 2 of order 2. [ ] ( ) We define representatives of our cohomology classes by constructing a low dimensional ap- proximation for a cellular chain complex of the universal cover of a K Sn, 1 -space. Furthermore, ( ) we show that the 2-chains of this chain complex are generated by the Sn orbits of three classes of elements: cij , dijkℓ, and eijk. Respectively, each of these three types of generators corresponds to the squaring, commuting, and braid relations in the presentation for the symmetric group. Therefore, the cocycle representing κ is determined as a function from the Sn module generated n 2 [ ] by these three 2-chains to Z2 . Using the presentation: n 2 2 Z2 = g¯ij 1 i j n g¯ = 1, gij ,gkℓ = 1 (1) ⟨{ } ≤ ≠ ≤ S ij [ ] ⟩ our second theorem defines the 2-cocycle classifying Zn. Theorem 1.2. If n ≥ 4, then a representative for the cocycle classifying Zn as an extension of n 2 Sn by Z2 is given by: κ cij = g¯ij ( ) ⎧g¯ik + g¯kj + g¯iℓ + g¯ℓj i < k < j < ℓ or k < i < ℓ < j κ dijkℓ = ⎪ ( ) ⎨0 otherwise ⎪ ⎩⎪ ⎧g¯ij + g¯jk i < k < j, j < i < k, or k < j < i κ eijk = ⎪ ( ) ⎨0 otherwise ⎪ ⎩⎪ In the construction of Theorem 1.2 we give a presentation for Zn. By considering Artin’s original presentation for the braid group given by: bibi+1bi = bi+1bibi+1 for all i Bn = b1,...,bn (2) d W bibj = bj bi if i − j > 1 i S S our presentation for Zn yields a normal generating set for Bn 4 . [ ] Theorem 1.3. Let i < k. As a subgroup of the braid group, Bn 4 is normally generated by elements of the form: [ ] 1. b2,b2 [ i k] 4 2. bi Outline This paper will begin with preliminaries which will define the conventions and def- initions for constructing group presentations, group cohomology, and braid/symmetric groups required. In section 3 we build an approximation for a low dimensional resolution for the univer- sal cover of a K Sn, 1 space and define the maps needed to construct the 2-cocycle using this ( ) 2 n Z2 resolution. Then in section 4 we construct the 2-cocycle φ ∈ Hθ Sn; and prove the order of this element. In section 5 we prove κ is determined by[φ], finishing( the proof) of Theorems 1.1 and 1.2. The proof of Theorem 1.3 is finished at the end of the paper with an explanation of the difficulties which arise in finding a finite generating set for Bn 4 . [ ] 2 1.1 Acknowledgments First I would like to give my deepest gratitude to my advisor Matthew Day for his continuous guidance, suggestions, and comments throughout this research and early drafts of this paper. I would also like to thank Dan Margalit and Matthew Clay for helpful conversations. 2 Preliminaries 2.1 Building a Group Presentation Given a group extension 1 Kι Gπ Q 1 and group presentations for K and Q, we describe the process to build a group presentation for G. This construction is previously known and included for completeness. Let SK RK and ⟨ S ⟩ S¯Q R¯Q be group presentations for K and Q respectively. For any set S, let F S denote the free⟨ groupS ⟩ generated by S. ( ) Generators: Since Q = G K, we may lift any elementg ¯ ∈ Q to an element g ∈ G by choosing ~ a g such that π g = g¯. Fix SQ ⊂ G as choice of lifts from elements in S¯Q such that π restricted ( ) to SQ is a bijection. Define S = SK ∪ SQ as the generators of G. Relations: Since K ≈ SK RK , any element k ∈ K can be considered as a word k1⋯kn ∈ ⟨ S ⟩ −1 F SK where ki ∈ SK for all i. Furthermore since K is normal in G, sks ∈ K for any s ∈ SQ and( k )∈ K; therefore we can choose k′ ∈ K such that sks−1 = k′ and we get a relation in G. Note ′ that for any k ∈ SK, k is determined by the choice of each s ∈ SQ as a lift of an element in S¯Q. ′ Let R be the relations in G defined by conjugating elements of SK by elements of SQ and their inverses. Now, we can write any relation in R¯Q bys ¯1⋯s¯n = 1Q wheres ¯i ∈ S¯Q for all i, therefore s1⋯sn = k for some k ∈ K. Note that the choices for si as a lifts ofs ¯i determines k. We take s1⋯sn = k in G as the lift of the relations ¯1⋯s¯n = 1Q in Q. Let RQ denote the lift of relations in ′ ′ R¯Q determined by the choice of SQ. Define R = RK ∪ RQ ∪ R , then G = S R . ⟨ S ⟩ Lemma 2.1. Let K, G′,Q be groups with presentations be defined as above. Any element g ∈ G′ can be written as g = w1w2 where w1 ∈ F SQ and w2 ∈ F SK . ( ) ( ) ′ Proof. Let g ∈ G , then g = s1⋯sn where si ∈ S. By induction on n, suppose g = s1s2. In this ′ case the statement is trivial unless s1 ∈ Sk and s2 ∈ SQ. By the relations of R (conjugating SK −1 by elements of SQ and the inverses) there exists k ∈ K such that s2 s1s2 = k. Consider −1 s1s2 = s2s2 s1s2 = s2k Since K is generated by SK , k = w2 for some w2 ∈ F SK and hence the Lemma holds for n = 2. Now, suppose the Lemma is true for any element( of) G′ that is a reduced word of length n − 1 in F S . Suppose g = s1⋯sn, then by induction s1⋯sn 1 = w1w2 where w1 ∈ F SQ and ( ) − ( ) w2 ∈ F SK .
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