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Generators for the mapping class group. (English) Zbl 0732.57004

Topology of low-dimensional manifolds, Proc. 2nd Sussex Conf., 1977, Lect. Notes Math. 722, 44-47 (1979).
[For the entire collection see Zbl 0397.00013).]
It is shown that there are \(2g+1\) simple closed curves on a closed orientable surface \(T_ g\) of genus g such that the isotopy classes of the Dehn twists about these curves generate the mapping class group \(M_ g\) of \(T_ g\). Moreover, this number is minimal, among all collections of Dehn twists which generate \(M_ g\). The proof of the first assertion proceeds from a known set of 3g-1 Dehn twists which generate \(M_ g\); it is shown that g-2 of those generators are isotopic to products of the remaining \(2g+1\) generators. The second assertion is established by showing that if twists about simple closed curves \(d_ 1,...,d_ n\) generate \(M_ g\), then \(d_ 1,...,d_ n\) span \(H_ 1(T_ g;{\mathbb{Z}}_ 2)\), hence \(n\geq 2g\). A separate argument shows that equality is impossible.

MSC:

57M20 Two-dimensional complexes (manifolds) (MSC2010)

Citations:

Zbl 0397.00013