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A survey of the Torelli group. (English) Zbl 0553.57002

Contemp. Math. 20, 165-179 (1983).
The mapping class group in genus g is the group of isotopy classes of orientation preserving homeomorphisms of a closed oriented surface of genus g. In variant forms one considers isotopy classes fixing a base point or of a surface with (connected) boundary. In any of these forms there is a homomorphism onto the symplectic group, Sp(2g), obtained by the action on the first homology group of the surface. The kernel of this homomorphism is known as the Torelli group in (genus g, and of the particular variant). This paper gives an informal, but informative, summary of the Torelli group - determining generators, describing its abelian quotients and other homogical information, explaining its connections to Teichmüller theory and 3-dimensional topology, and in the process raising many open questions.
For the entire collection see [Zbl 0512.00022].
Reviewer: J.Hempel

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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