Concept

Cross-cap

A cross-cap is a Möbius band sewn into a hole cut in a surface, the one-sided counterpart of a handle. Every one-sided closed surface is a sphere with some number of cross-caps, and each one lowers the Euler characteristic by one.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The gluing abab makes a projective plane. A polygon whose edges carry the word abab, with arrows for the direction each edge is glued and the corners coloured by which vertex they become.

A disc sewn to a Möbius band

The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.

topology · Orientability

The two-sided cover of a sphere with 3 cross-caps, built from aabbcc. Two copies of the polygon aabbcc with opposite senses of turning, glued within or across copies; 2 vertices, 6 edges, 2 faces, characteristic −2: a surface with 2 handles.

Two copies of the polygon, cross-matched

The two-sided surface that lies over a one-sided one can be built from the gluing word alone: take two copies of the polygon, one read each way round, glue a letter within the copies when it is used once each way, and across them when it is used twice the same way. The recipe works for every surface at once, doubles every count, and shows that a sphere with k cross-caps is covered by the surface with k − 1 handles — and that among all its connected double covers, exactly one is two-sided.

topology · Orientability

Named alongside it

The objects these essays reach for when they reach for this one.

Classification Euler characteristic Gluing diagram Orientability Antipodal map Closed surface Connected sum Covering space Immersion Non-orientable Orientation double cover Projective plane

All concepts