Crosscap slides and the level 2 mapping class group of a nonorientable surface

Abstract

Crosscap slide is a homeomorphism of a nonorientable surface of genus at least 2, which was introduced under the name Y-homeomorphism by Lickorish as an example of an element of the mapping class group which cannot be expressed as a product of Dehn twists. We prove that the subgroup of the mapping class group of a closed nonorientable surface N generated by all crosscap slides is equal to the level 2 subgroup consisting of those mapping classes which act trivially on H 1 (N ; Z 2 ). We also prove that this subgroup is generated by involutions.

A normal generating set for the Torelli group of a non-orientable closed surface

“…The black cross "×" means the composition of t c 1 with t c 2 . By the proof of Lemma 3.1 in[17], we have Y −1 j;i Y…”

“…On generators for Γ 2 (N g ). McCarthy, Pinkall [11] and Gadgil, Pancholi [5] proved that the natural homomorphism [17] proved that Γ 2 (N g ) is generated by Y -homeomorphisms, and that Γ 2 (N g ) is generated by involutions. Therefore,…”

On p–almost direct products and residual properties of pure braid groups of nonorientable surfaces

“…In this section, we recall the construction of the Blowup homomorphism η n g,b : Γ n g,b −→ Γ g+n,b given in [Sz1], [Sz2] and [PS].…”

“…Such a homeomorphism h commutes with the identification leading to N g+n,b and thus induces an element η(h) ∈ Γ g+n,b . It is proved in [Sz2] that the map η n g,b = η : Γ n g,b −→ Γ g+n,b which sends h to η(h) is well defined for n = 1, but the proof also works for n > 1. This homomorphism is called blowup homomorphism.…”

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