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

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

So, the y-homeomorphism is defined by a isotopy leaving the boundary fixed by sliding the second band one turn around the core of till the original position. The result is an automorphism of which maps 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.

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