Concept

Orientation double cover

The two-sided surface that lies two to one over a surface, each of its points being a point below together with a choice of sense of turning there. It is connected exactly when the surface below is one-sided, and it doubles the Euler characteristic.

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

A sphere over a projective plane, cell by cell. An icosahedron with opposite faces drawn in matching colours, beside the count of cells it has and the count the quotient by the antipodal map has — every number halved, including the Euler characteristic.

Two sheets over a one-sided surface

Above every one-sided surface sits a two-sided one, exactly twice as large, and the map between them forgets which of the two senses of turning a point was carrying. Building it turns a question about sides into a question about covers.

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.

Covering space Euler characteristic Orientability Antipodal map Classification Cross-cap Deck transformation Free action Fundamental group Gluing diagram Non-orientable Projective plane

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