## Geometry & Topology

### An elementary approach to the mapping class group of a surface

Bronisław Wajnryb

#### Abstract

We consider an oriented surface $S$ and a cellular complex $X$ of curves on $S$, defined by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex $X$ is connected and simply connected. From this we derive an explicit simple presentation of the mapping class group of $S$, following the ideas of Hatcher–Thurston and Harer.

#### Article information

Source
Geom. Topol., Volume 3, Number 1 (1999), 405-466.

Dates
Revised: 18 November 1999
Accepted: 6 December 1999
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883153

Digital Object Identifier
doi:10.2140/gt.1999.3.405

Mathematical Reviews number (MathSciNet)
MR1726532

Zentralblatt MATH identifier
0947.57015

#### Citation

Wajnryb, Bronisław. An elementary approach to the mapping class group of a surface. Geom. Topol. 3 (1999), no. 1, 405--466. doi:10.2140/gt.1999.3.405. https://projecteuclid.org/euclid.gt/1513883153

#### References

• J S Birman, H Hilden, On mapping class groups of closed surfaces as covering spaces, from: “Advances in the theory of Riemann surfaces”, Ann. Math. Stud. 66 (1971) 81–115
• J S Birman, Braids, links and mapping class groups, Annals of Math. Studies 82 (1974)
• J S Birman, B Wajnryb, Errata: presentations of the mapping class group, Israel J. Math. 88 (1994) 425–427
• J Cerf, La stratification naturelle \dots, Publ. Math. I.H.E.S. 39 (1970) 5–173
• M Dehn, Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938) 135–206
• S Gervais, A finite presentation of the mapping class group of an oriented surface, preprint
• J Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983) 221–239
• J Hass, P Scott, Intersections of curves on surfaces, Israel J. Math. 51 (1985),90–120.
• A Hatcher, W Thurston, A presentation for the mapping class group of a closed orientable surface, Topology, 19 (1980) 221–237
• M Heusner, Eine Präsentation der Abbildungsklassengruppe einer geschlossenen, orientierbaren Fläche, Diplom-Arbeit, University of Frankfurt
• S Humphries, Generators for the mapping class group, from: “Topology of Low-dimensional Manifolds”, LNM 722 (1979) 44–47
• D Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979) 119–125
• R C Kirby, A calculus of framed links in $S^3$, Invent. Math. 45 (1978) 35–56
• T Kohno, Topological invariants for 3–manifolds using representations of mapping class groups I, Topology, 31 (1992) 203–230
• F Laudenbach, Presentation du groupe de diffeotopies d'une surface compacte orientable, Travaux de Thurston sur les surfaces, Asterisque 66–67 (1979) 267–282
• Ning Lu, A simple proof of the fundamental theorem of Kirby calculus on links, Trans. Amer. Math. Soc. 331 (1992) 143–156
• M Matsumoto, A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities, preprint
• S Matveev, M Polyak, On a tangle presentation of the mapping class groups of surfaces, Contemporary Mathematics, 164 (1994) 219–229
• J McCool, Some finitely presented subgroups of the automorphism group of a free group, J. Algebra, 35 (1975) 205–213
• B Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983) 157–174