Concept

Lifting

Following a path or a map upstairs through a covering, so that it projects back to the original. A path lifts uniquely once a starting point above is chosen, and whether a loop lifts to a loop is what decides which subgroup a cover names.

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

3 sheets, and the subgroup they name. A circle with its 3-sheeted cover drawn as a spiral above it, beside a table of the winding classes and whether each lifts to a closed loop. The ones that do are exactly the multiples of 3.

Every cover is a subgroup

A space can be unrolled, and the ways of unrolling it are not arbitrary. They correspond exactly to the subgroups of its fundamental group — index equals sheets, normality equals symmetry — so a question about a group becomes a question about a picture and back again.

topology · Homotopy

Named alongside it

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

Covering space Fundamental group Homotopy Winding number Deck transformation Group Index Loop Orientation Subgroup

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