Generating the surface mapping class group by two elements

Wajnryb proved in 1996 that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the mapping class groups are also generated by two elements of finite order.


Relative flux homomorphism in symplectic geometry
Ozan, Yıldıray (American Mathematical Society (AMS), 2005-01-01)
In this work we define a relative version of the flux homomorphism, introduced by Calabi in 1969, for a symplectic manifold. We use it to study ( the universal cover of) the group of symplectomorphisms of a symplectic manifold leaving a Lagrangian submanifold invariant. We also show that some quotients of the universal covering of the group of symplectomorphisms are stable under symplectic reduction.
Arıkan, Mehmet Fırat; Salur, Sema (International Press of Boston, 2013-06-01)
In this paper, we show the existence of (co-oriented) contact structures on certain classes of G(2)-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with G(2)-structure) admits an almost contact structure. We also construct explicit almost contact metric structures on manifolds with G(2)-structures.
On symplectic quotients of K3 surfaces
Cinkir, Z; Onsiper, H (Elsevier BV, 2000-12-18)
In this note, we construct generalized Shioda-Inose structures on K3 surfaces using cyclic covers and almost functoriality of Shioda-Inose structures with respect to normal subgroups of a given group of symplectic automorphisms.
Legendrian realization in convex Lefschetz fibrations and convex stabilizations
Akbulut, Selman; Arıkan, Mehmet Fırat (Walter de Gruyter GmbH, 2015-05-01)
We show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on W, any Lagrangian submanifold with trivial first de Rham cohomology group, embedded on a (symplectic) page of the (induced) convex open book on partial derivative W, can be assumed to be Legendrian in partial derivative W with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. We also s...
The second homology groups of mapping class groups of orientable surfaces
Korkmaz, Mustafa (Cambridge University Press (CUP), 2003-05-01)
Let $\Sigma_{g,r}^n$ be a connected orientable surface of genus $g$ with $r$ boundary components and $n$ punctures and let $\Gamma_{g,r}^n$ denote the mapping class group of $\Sigma_{g,r}^n$, namely the group of isotopy classes of orientation-preserving diffeomorphisms of $\Sigma_{g,r}^n$ which are the identity on the boundary and on the punctures. Here, we see the punctures on the surface as distinguished points. The isotopies are required to be the identity on the boundary and on the punctures. If $r$ and...
Citation Formats
M. Korkmaz, “Generating the surface mapping class group by two elements,” TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, pp. 3299–3310, 2005, Accessed: 00, 2020. [Online]. Available: