## Geometry & Topology

### Cutting and pasting in the Torelli group

Andrew Putman

#### Abstract

We introduce machinery to allow “cut-and-paste”-style inductive arguments in the Torelli subgroup of the mapping class group. In the past these arguments have been problematic because restricting the Torelli group to subsurfaces gives different groups depending on how the subsurfaces are embedded. We define a category $TSur$ whose objects are surfaces together with a decoration restricting how they can be embedded into larger surfaces and whose morphisms are embeddings which respect the decoration. There is a natural “Torelli functor” on this category which extends the usual definition of the Torelli group on a closed surface. Additionally, we prove an analogue of the Birman exact sequence for the Torelli groups of surfaces with boundary and use the action of the Torelli group on the complex of curves to find generators for the Torelli group. For genus $g≥1$ only twists about (certain) separating curves and bounding pairs are needed, while for genus $g=0$ a new type of generator (a “commutator of a simply intersecting pair”) is needed. As a special case, our methods provide a new, more conceptual proof of the classical result of Birman and Powell which says that the Torelli group on a closed surface is generated by twists about separating curves and bounding pairs.

#### Article information

Source
Geom. Topol., Volume 11, Number 2 (2007), 829-865.

Dates
Accepted: 10 April 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799862

Digital Object Identifier
doi:10.2140/gt.2007.11.829

Mathematical Reviews number (MathSciNet)
MR2302503

Zentralblatt MATH identifier
1157.57010

#### Citation

Putman, Andrew. Cutting and pasting in the Torelli group. Geom. Topol. 11 (2007), no. 2, 829--865. doi:10.2140/gt.2007.11.829. https://projecteuclid.org/euclid.gt/1513799862

#### References

• M A Armstrong, On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965) 639–646
• J S Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–238
• J S Birman, On Siegel's modular group, Math. Ann. 191 (1971) 59–68
• J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press (1974)
• S Gervais, A finite presentation of the mapping class group of a punctured surface, Topology 40 (2001) 703–725
• P Gold, On the mapping class and symplectic modular group, PhD thesis, New York University (1961)
• A J Hahn, O T O'Meara, The classical groups and $K$-theory, with a foreword by J Dieudonné, Grundlehren series 291, Springer (1989)
• R M Hain, Torelli groups and geometry of moduli spaces of curves, from: “Current topics in complex algebraic geometry (Berkeley, CA, 1992/93)”, Math. Sci. Res. Inst. Publ. 28, Cambridge Univ. Press (1995) 97–143
• J L Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. $(2)$ 121 (1985) 215–249
• W J Harvey, Geometric structure of surface mapping class groups, from: “Homological group theory (Proc. Sympos., Durham, 1977)”, (C T C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 255–269
• N V Ivanov, Complexes of curves and Teichmüller modular groups, Uspekhi Mat. Nauk 42 (1987) 49–91, 255
• N V Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs 115, Amer. Math. Soc. (1992) Translated from the Russian by E J F Primrose and revised by the author
• D L Johnson, A survey of the Torelli group, from: “Low-dimensional topology (San Francisco, CA, 1981)”, Contemp. Math. 20, Amer. Math. Soc. (1983) 165–179
• D L Johnson, The structure of the Torelli group. I. A finite set of generators for ${\cal I}$, Ann. of Math. $(2)$ 118 (1983) 423–442
• D L Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology 24 (1985) 113–126
• D L Johnson, The structure of the Torelli group. III. The abelianization of $\mathcal T$, Topology 24 (1985) 127–144
• H Klingen, Charakterisierung der Siegelschen Modulgruppe durch ein endliches System definierender Relationen, Math. Ann. 144 (1961) 64–82
• W Magnus, Über $n$-dimensionale Gittertransormationen, Acta. Math. 64 (1934) 353–367
• J Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978) 347–350
• B van den Berg, On the Abelianization of the Torelli group, PhD thesis, University of Utrecht (2003)