Concept

Gluing

Building a space by identifying parts of simpler pieces, such as pairing the edges of a polygon so that each pair becomes one seam. The pairing word fixes the result up to topology, and every closed surface arises from a single polygon in this way.

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

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

The two-handled surface's octagon, eight to a corner. Regular hyperbolic octagon with angles π/4 and 64 neighbouring copies in the Poincaré disc.

An octagon only the hyperbolic plane can hold

Glue an octagon's sides in the pattern that makes a surface with two handles and its eight corners become one point, so their angles must add to a single turn. A flat octagon's add to three. The octagon that fits has corners of 45° and lives in the hyperbolic plane, where its area is forced to be exactly 4π and its four gluing moves are motions of the plane whose commutators multiply to the identity.

topology · Surface classification

Named alongside it

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

Fundamental group Curvature Euler characteristic Free group Generator Genus Group presentation Homotopy Hyperbolic geometry Mobius transformation Relation Surface

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