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Concept

Monodromy

The permutation of a covering's sheets obtained by following a loop in the base and noting which sheet each lift ends on. It records the whole covering as an action of the loops, and a branch point is a point round which it is not the identity.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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 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.

topology · Covering spaces 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.

topology · Covering spaces 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.

topology · Covering spaces Every failed branch datum up to eight sheets. Exceptions with three branch points: 4: 31|22|22; 6: 51|222|222; 6: 42|411|222; 6: 42|222|222; 6: 33|33|3111; 6: 33|321|222; 8: 71|2222|2222; 8: 62|5111|2222; 8: 62|2222|2222; 8: 611|422|2222; 8: 53|3311|2222; 8: 53|2222|2222; 8: 521|422|2222; 8: 5111|44|2222; 8: 5111|422|422; 8: 44|44|311111; 8: 44|3221|2222; 8: 431|332|2222; 8: 422|332|2222.

The lists no covering can have

A covering of the sphere by itself with three branch points is three permutations of the sheets whose product is the identity. The Riemann–Hurwitz count says which lists of cycle shapes might work, and a search through every permutation says which do. Up to thirteen sheets, 150 lists pass the count and fail the search — and not one of them has a prime number of sheets.

topology · Covering spaces

Named alongside it

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

Covering spaceEuler characteristicPermutationExistence proofFundamental groupGenusGroup actionSubgroupConjugacy classCounting argumentExhaustive searchFree group

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