y-homeomorphism

The y-homeomorphism also dubbed crosscap slide, is an auto-homeomorphism (or self-homeomorphism) which can be defined only fornon 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 aprojective 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 thecore 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 thehomeotopy group of a non-orientable surface, Proc. Camb. Phil. Soc.65(1969), 409-430.
2.
M. Korkmaz. Mapping Class Groups of Non-orientable Surfaces, Geometriae Dedicata 89 (2002), 109-133.
3.
W.B.R. Lickorish. Homeomorphisms of non-orientable two-manifolds,Math. Proc. Camb. Phil. Soc. 59 (1963), 307-317.