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The structure of the Torelli group. III: The abelianization of \({\mathcal S}\). (English) Zbl 0571.57010
The author continues his profound study of the Torelli group \({\mathcal T}\), that is the subgroup of the mapping class group M of a compact surface which acts trivially on homology.
From the introduction: ”In the first paper [Ann. Math., II. Ser. 113, 423-442 (1983; Zbl 0549.57006)] we treated the problem of finite generation of \({\mathcal T}\); the ideas and notation of that paper reoccur here, and we assume familiarity with them. In this paper we calculate two abelian quotients of \({\mathcal T}\), namely the universal abelian quotient \({\mathcal T}/{\mathcal T}'\) and the universal \({\mathbb{Z}}_ 2\)-vector space quotient \({\mathcal T}/{\mathcal T}^ 2\), where \({\mathcal T}^ 2\) is the subgroup generated by all squares in \({\mathcal T}\). If \(M_{g,1}\) is a surface of genus \(g\geq 3\) with one boundary component and \({\mathcal T}_{g,1}\) is its Torelli group, then the author has previously constructed two surjective homomorphisms \(\sigma\) : \({\mathcal T}_{g,1}/{\mathcal T}^ 2_{g,1}\to B^ 3_{g,1}\), where the target group is a certain \({\mathbb{Z}}_ 2\)-vector space of cubic polynomials and \(\tau\) : \({\mathcal T}_{g,1}/{\mathcal T}'\!_{g,1}\to \Lambda^ 3H_ 1(M_{g,1},{\mathbb{Z}})\) where the target is the 3rd exterior power of the homology of \(M_{g,1}.\)
The principal results are as follows: (a) \(\sigma\) is an isomorphism, and hence, \({\mathcal T}^ 2=[{\mathcal M}^{(2)},{\mathcal T}]={\mathcal C}\), where \({\mathcal M}^{(2)}\) is the subgroup of \({\mathcal M}\) which acts trivially on \(H_ 1(M_{g,1},{\mathbb{Z}}_ 2)\) and \({\mathcal C}\) is the common kernel of all the Birman-Craggs homomorphims [J. S. Birman and R. Craggs, Trans. Am. Math. Soc. 237, 283-309 (1978; Zbl 0383.57006)]. (b) \({\mathcal T}'\!_{g,1}=Ker \sigma \cap Ker \tau\), and hence \({\mathcal T}_{g,1}/{\mathcal T}'\!_{g,1}\) is a certain pullback constructed from \(B^ 3_{g,1}\) and \(\Lambda^ 3H_ 1(M_{g,1},{\mathbb{Z}}).''\)
Reviewer: B.Zimmermann

MSC:
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M99 General low-dimensional topology
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57R50 Differential topological aspects of diffeomorphisms
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