Mapping class groups of nonorientable surfaces

We obtain a finite set of generators for the mapping class group of a nonorientable surface with punctures. We then compute the first homology group of the mapping class group and certain subgroups of it. As an application we prove that the image of a homomorphism from the mapping class group of a nonorientable surface of genus at least nine to the group of real-analytic diffeomorphisms of the circle is either trivial or of order two.


Automorphisms of curve complexes on nonorientable surfaces
Atalan, Ferihe; Korkmaz, Mustafa (2014-01-01)
For a compact connected nonorientable surface N of genus g with n boundary components, we prove that the natural map from the mapping class group of N to the automorphism group of the curve complex of N is an isomorphism provided that g + n >= 5. We also prove that two curve complexes are isomorphic if and only if the underlying surfaces are diffeomorphic.
Generating the Mapping Class Group of a Nonorientable Surface by Two Elements or by Three Involutions
Altunöz, Tülin; Pamuk, Mehmetcik; Yildiz, Oguz (2022-01-01)
We prove that, for g≥ 19 the mapping class group of a nonorientable surface of genus g, Mod (Ng) , can be generated by two elements, one of which is of order g. We also prove that for g≥ 26 , Mod (Ng) can be generated by three involutions.
Generating the twist subgroup by involutions
Altunöz, Tülin; Pamuk, Mehmetcik; Yildiz, Oguz (2020-01-01)
For a nonorientable surface, the twist subgroup is an index 2 subgroup of the mapping class group generated by Dehn twists about two-sided simple closed curves. In this paper, we consider involution generators of the twist subgroup and give generating sets of involutions with smaller number of generators than the ones known in the literature using new techniques for finding involution generators.
A note on the generalized Matsumoto relation
DALYAN, ELİF; Medetogullari, Elif; Pamuk, Mehmetcik (2017-01-01)
We give an elementary proof of a relation, first discovered in its full generality by Korkmaz, in the mapping class group of a closed orientable surface. Our proof uses only the well-known relations between Dehn twists.
Reference-plane-invariant waveguide method for electromagnetic characterization of bi-axial bianisotropic metamaterials
HASAR, UĞUR CEM; Yildiz, Gul; BUTE, MUSA; Muratoğlu, Abdurrahim (2018-11-01)
In this paper, we investigate a reference-plane invariant (RPI) method for electromagnetic property extraction of bi-axial bianisotropic metamaterial (MM) slabs. In order to obtain unique properties, we applied the frequency varying technique in order to determine the location of the slab within its cell. For validation of the proposed method, we first simulated and then measured scattering parameters of a MM slab constructed by split-ring-resonators, next extracted its electromagnetic properties, and final...
Citation Formats
M. Korkmaz, “Mapping class groups of nonorientable surfaces,” GEOMETRIAE DEDICATA, pp. 109–133, 2002, Accessed: 00, 2020. [Online]. Available: