The structure of the Torelli group. II: A characterization of the group generated by twists on bounding curves.

*(English)*Zbl 0571.57009This 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