Matpic
mathematics, in pictures
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 4 essays across 2 fields — 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 3 sheets, 8 of 26 words coming back. A table of reduced words in two generators with the sheet each sends the base sheet to. The words returning to it are the covering's subgroup, and the 3 sheets are its cosets.

A covering is a permutation

Describing a covering means saying where each loop sends each sheet, which is a permutation for every generator. So a covering of a wedge of circles is nothing but a homomorphism to a symmetric group, and the subgroup it corresponds to is a stabiliser.

topology · Covering spaces Square roots of 2, lifted from one power of 7 to the next. A tree whose rows are the square roots of 2 modulo 7 to the powers 1 to 4, each joined to the root it reduces to; counts 2, 2, 2, 2.

A root lifted one digit at a time

On a dial of seven, 3 × 3 is 2. On a dial of forty-nine the square root of 2 must reduce to 3, so there are only seven candidates, and exactly one of them works: 10. On a dial of 343 exactly one lift of 10 works: 108. Each step adds one digit on the left, found by solving a linear equation, and the digits go on for ever — a number …21216213 whose square is 2, in a world where closeness means divisibility by seven.

discrete · Modular arithmetic

Named alongside it

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

Covering spaceFundamental groupHomotopyIndexSubgroupWinding numberDeck transformationDerivativeFree groupGroupGroup actionHensel lemma

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