Concept

Base point

The point at which loops in a space are required to start and finish, so that one can be run after another. Changing it gives an isomorphic fundamental group, but only up to conjugation by the path used, which is why base-point-free statements are about conjugacy classes.

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

The same loop, seen from two places. An annulus with two marked points and a loop based at the first. Two paths join the points, differing by a full turn round the hole, and each carries the loop to a loop based at the second point.

The group a space has at a point

The loops of a space form a group once a starting point is fixed, and the fixing looks like an arbitrary choice that ought to be removable. It is removable, but only up to conjugation, and the residue is exactly what makes a non-commutative fundamental group harder to state than to compute.

topology · Homotopy

Sliding one square past another. 4 stages of a slide in which two labelled squares inside a larger one exchange positions without ever overlapping. The larger square's boundary is the base point throughout, and the exchange is what makes the composition commutative.

Why the second group commutes

Replace loops by spheres and the same construction gives a second homotopy group. It is always commutative, and the reason is not a fact about spheres or about any space — it is a two-line argument about any set carrying two compatible operations.

topology · Homotopy

Named alongside it

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

Fundamental group Group Homotopy Commutativity Conjugacy Eckmann hilton Equivalence relation Higher homotopy Path connected Sphere Winding number

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