Concept

Deck transformation

A symmetry of a covering space that leaves the covering map unchanged, permuting the sheets over each point. The collection of them is a group whose size is the number of sheets, and it records what the covering remembers that the base has forgotten.

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

The line spiralling over the circle. A circle with a helix drawn above it: the helix is the real line, and the map that sends each of its points straight down onto the circle covers the circle once per turn. Above one marked point sits a column of points, one per turn.

The same loop, unrolled

Spread a circle out into a line spiralling above it, and a loop that closes downstairs becomes a path that does not — so a question about which loops can be shrunk becomes a question about where a path ends, which is easy.

topology · covering spaces

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

Named alongside it

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

Covering space Fundamental group Antipodal map Euler characteristic Free action Homotopy Lifting Loop Orientability Orientation Orientation double cover Projective plane

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