Concept

Covering space

A space mapping onto another so that each point below has a neighbourhood whose preimage is a stack of separate copies. Paths lift to it uniquely, which turns a question about which loops shrink into a question about where a lifted path ends.

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

Also named here as fundamental group, homotopy, loop — the same set of essays touches all of them, so they are one junction rather than several.

3 loops in one ring, and the number that separates them. Loops drawn in an annulus, each labelled with how many times it goes round the hole. Loops with different counts cannot be deformed into one another without leaving the ring.

A loop that cannot be pulled tight

A hole is a strange thing to point at, because it is precisely where the surface is not. What can be pointed at is a loop of string lying on the surface — and the hole announces itself by refusing to let that loop be pulled in to a point.

topology · homotopy

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

Named alongside it

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

Fundamental group Homotopy Loop Winding number Continuity Deck transformation Deformation Genus Invariant Lifting Orientation Topological invariant

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