y-homeomorphism

The also dubbed crosscap slide, is an auto-homeomorphism (or self-homeomorphism) which can be defined only for non orientable surfaces whose genus is greater than one.

To define it we take a punctured Klein bottle $K_{0}=K\setminus{\rm int}\ D^{2}$ 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:

 $K_{0}=(M_{1}\setminus{\rm int}\ D^{2})\cup_{\partial}M_{2}$

where $\partial=\partial D^{2}=\partial M_{2}$. Other way to visualizing that, is by consider $K_{0}$ as the connected sum   of ${\rm int}\ M_{1}$ 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 y-homeomorphism 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. 1.

D.R.J. Chillingworth. , Proc. Camb. Phil. Soc. 65(1969), 409-430.

2. 2.

M. Korkmaz. , Geometriae Dedicata 89 (2002), 109-133.

3. 3.

W.B.R. Lickorish. , Math. Proc. Camb. Phil. Soc. 59 (1963), 307-317.

Title y-homeomorphism Yhomeomorphism 2013-03-22 15:42:26 2013-03-22 15:42:26 juanman (12619) juanman (12619) 8 juanman (12619) Definition msc 54C10 crosscap slide CrosscapSlide