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The structure of the Torelli group. II: A characterization of the group generated by twists on bounding curves. (English) Zbl 0571.57009
This is the second of three of the author’s papers concerning the Torelli-group $${\mathcal S}_{g,n}$$ of a compact surface $$M=M_{g,n}$$ (of genus g with n boundary components). [Part I, Ann. Math., II. Ser. 118, 423-442 (1983; Zbl 0549.57006) and Contemp. Math. 20, 165-179 (1983; Zbl 0553.57002)]. By definition, this is the subgroup of the mapping class group of M acting trivially on homology $$H_ 1(M,{\mathbb{Z}})$$. It is generated by Dehn twists on bounding simple closed curves on M and opposite Dehn twists on pairs of homologous disjoint simple closed curves. Let $${\mathcal T}$$ be the subgroup of $${\mathcal S}$$ generated by all generators of the first type. In the first paper the author showed that $${\mathcal S}_{g,0}$$ and $${\mathcal S}_{g,1}$$ are finitely generated, for $$g\geq 3$$ (in fact, finitely many of the above twist-generators suffice). In still another paper the author constructed a surjective homomorphism $$\tau$$ : $${\mathcal S}_{g,1}\to \Lambda^ 3H_ 1(M_{g,1},{\mathbb{Z}})$$ (third exterior power), with $${\mathcal T}\subset \ker nel$$ $$\tau$$, and conjectured that $${\mathcal T}=\ker nel \tau$$. The proof of this conjecture is the content of the present second paper. In the third paper the results of the second are used to compute the abelianization $${\mathcal S}/{\mathcal S}^ 1$$.
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
##### Keywords:
Torelli-group; compact surface; mapping class group; Dehn twists
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