The y-homeomorphismPlanetmathPlanetmath 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 K0=KintD2 which can be consider as a pair of closed Möbius bands M1,M2, one sewed in the other by perforating with a disk (being disjoint from M1) and then identify the boundary of the second with the boundary of that disk, in symbols:


where =D2=M2. Other way to visualizing that, is by consider K0 as the connected sumMathworldPlanetmathPlanetmath of intM1 with a projective planeMathworldPlanetmath P2.

Now, thinking that the removed disk D2 was located with its center at some point in the core of M1, the second band, M2 will have a pair of points on that part of the core in common with M2.

So, the y-homeomorphism is defined by a isotopyPlanetmathPlanetmath leaving the boundary M1 fixed by sliding the second band M2 one turn around the core of M1 till the original position. The result is an automorphismMathworldPlanetmathPlanetmathPlanetmath of K0 which maps M2 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. A finite set of generatorsPlanetmathPlanetmathPlanetmath for the homeotopy group of a non-orientable surface, Proc. Camb. Phil. Soc. 65(1969), 409-430.

  2. 2.

    M. Korkmaz. Mapping Class GroupsPlanetmathPlanetmath of Non-orientable Surfaces, Geometriae Dedicata 89 (2002), 109-133.

  3. 3.

    W.B.R. Lickorish. HomeomorphismsMathworldPlanetmath of non-orientable two-manifolds, Math. Proc. Camb. Phil. Soc. 59 (1963), 307-317.

Title y-homeomorphism
Canonical name Yhomeomorphism
Date of creation 2013-03-22 15:42:26
Last modified on 2013-03-22 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