Covering spaces — the series

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.

part 1 · topology

A wedge of 2 circles. Several circles all passing through one common point, each labelled with a generator, so that a loop is a word in those letters.

The subgroup that is freer than the group

A free group on two letters contains a subgroup of index three that is free on four. Nothing about a group makes that plausible; everything about a graph makes it obvious, and the argument is to stop looking at the group and start looking at the space whose loops it is.

part 2 · topology

3 symmetries over 3 sheets: a regular covering. A 3-sheeted covering of a wedge of 2 circles, with the permutations of its sheets that commute with every generator. There are 3, against 3 sheets.

The symmetries a cover has of its own

A covering space can be shuffled without disturbing anything below it, and how many ways there are is decided by the subgroup it corresponds to. When there are as many symmetries as sheets the covering is called regular, and that is the same statement as the subgroup being normal.

part 3 · topology

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.

part 4 · topology

6 vertices folded to 4, and a graph that decides. The graph built from 3 generator words, folded until no vertex has two edges of one label leaving it. Reading a word from the base vertex decides membership, and 6 words are tested.

Folding a graph until it decides

A subgroup of a free group usually arrives as a list of words, and almost nothing about it is readable from the list. Draw the words as loops, merge every pair of edges with the same label leaving one point, and what is left is a machine that decides membership by reading.

part 5 · topology

3 sheets over a surface of genus 2: a surface of genus 4. A 3-sheeted covering of the closed surface of genus 2, drawn as 3 copies of its 8-sided face with each side coloured by its generator and numbered with the sheet it glues to. The Euler characteristic −6 is 3 times −2, and the cover has genus 4.

Covering a surface multiplies its count

A covering of a closed surface is a permutation of the sheets for each edge of the surface's one face — with one condition that a covering of a graph never had to meet. When the condition holds, the cells of the cover can be counted directly, and the count is the base's count times the number of sheets. That multiplication decides which surfaces can cover which, before any cover is built.

part 6 · topology

2 sheets branched over 4 points of a sphere: a surface of genus 1. A 2-sheeted branched covering of the sphere with 4 branch points, drawn as 2 rows of sheets over a centre and the branch points, with the sheets joined where each point's permutation cycles them. Counting cells gives Euler characteristic 0, matching the Riemann–Hurwitz formula, and genus 1.

What a branch point subtracts

Let the sheets of a covering meet at a few points and the count stops multiplying — but it fails by an amount that can be read off each point's permutation. Cut the sphere into a star, lift the cells, and the Riemann–Hurwitz formula falls out of a subtraction. The same count then turns out to be necessary and not sufficient.

part 7 · topology

Six lists of cycle shapes, and how many coverings each has. A table of lists of cycle shapes over a sphere, each with the Euler characteristic the Riemann–Hurwitz count gives, the number of lists of permutations with that product, and the number of those that connect all the sheets.

A count that can say zero

The branched count ends on a list of cycle shapes that passes every test and describes no covering. There is an exact formula for how many coverings a list has — a sum over the character table of a symmetric group — and it returns nought without giving any reason why.

part 8 · topology