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mathematics, in pictures
Concept

Generator

An element of a group from which, together with its fellows, every other element can be built by multiplication and inversion. A group given by generators and relations is presented rather than described, and almost every question about it is then a question about words.

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

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.

topology · Covering spaces Three spaces, cut into pieces. 3 panels, drawn from: a sphere split into two caps meeting along a circle, two circles joined at a point, and a square whose opposite edges are glued into a torus. Each carries the fundamental group that van Kampen's theorem computes for it.

Cutting a space to find its group

A space assembled from two pieces has a fundamental group assembled from theirs, and the recipe is exact — take everything both groups offer and impose the relations the overlap forces. Almost every fundamental group anybody knows is computed this way, including all of the surfaces.

topology · Homotopy

Named alongside it

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

Free groupFundamental groupCovering spaceGluingGraphGroup presentationHomotopyIndexLoopRankRelationSpanning tree

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