yhomeomorphism
The yhomeomorphism^{} also dubbed crosscap slide, is an autohomeomorphism (or selfhomeomorphism) which can be defined only for non orientable surfaces whose genus is greater than one.
To define it we take a punctured Klein bottle ${K}_{0}=K\setminus \mathrm{int}{D}^{2}$ which can be consider as a pair of closed Möbius bands ${M}_{1},{M}_{2}$, one sewed in the other by perforating with a disk (being disjoint from $\partial {M}_{1}$) and then identify the boundary of the second with the boundary of that disk, in symbols:
$${K}_{0}=({M}_{1}\setminus \mathrm{int}{D}^{2}){\cup}_{\partial}{M}_{2}$$ 
where $\partial =\partial {D}^{2}=\partial {M}_{2}$. Other way to visualizing that, is by consider ${K}_{0}$ as the connected sum^{} of $\mathrm{int}{M}_{1}$ with a projective plane^{} $\mathbb{R}{P}^{2}$.
Now, thinking that the removed disk ${D}^{2}$ was located with its center at some point in the core of ${M}_{1}$, the second band, ${M}_{2}$ will have a pair of points on that part of the core in common with $\partial {M}_{2}$.
So, the yhomeomorphism is defined by a isotopy^{} leaving the boundary $\partial {M}_{1}$ fixed by sliding the second band ${M}_{2}$ one turn around the core of ${M}_{1}$ till the original position. The result is an automorphism^{} of ${K}_{0}$ which maps ${M}_{2}$ into itself but reversing it.
To define this for genus greater than two just consider any other non orientable surface as a connected sum of a Kein bottle plus projective planes.

1.
D.R.J. Chillingworth. A finite set of generators^{} for the homeotopy group of a nonorientable surface, Proc. Camb. Phil. Soc. 65(1969), 409430.

2.
M. Korkmaz. Mapping Class Groups^{} of Nonorientable Surfaces, Geometriae Dedicata 89 (2002), 109133.

3.
W.B.R. Lickorish. Homeomorphisms^{} of nonorientable twomanifolds, Math. Proc. Camb. Phil. Soc. 59 (1963), 307317.
Title  yhomeomorphism 

Canonical name  Yhomeomorphism 
Date of creation  20130322 15:42:26 
Last modified on  20130322 15:42:26 
Owner  juanman (12619) 
Last modified by  juanman (12619) 
Numerical id  8 
Author  juanman (12619) 
Entry type  Definition 
Classification  msc 54C10 
Synonym  crosscap slide 
Related topic  CrosscapSlide 