Homotopy and the fundamental group

Builds on

Required prerequisites, all the way down: 40

You will learn to

• Define homotopy and the fundamental group, and prove that the fundamental group is a homotopy invariant.
• Compute π₁(S¹) ≅ ℤ using covering spaces and deduce the Brouwer fixed-point theorem for the disc.
• Use the Seifert–van Kampen theorem to compute fundamental groups of wedges of circles and of surfaces.

Definition

• A homotopy between maps f, g : X → Y is a continuous map H : X × [0, 1] → Y with H(x, 0) = f(x) and H(x, 1) = g(x).
• π₁(X, x₀) is the group of homotopy classes of loops based at x₀, with concatenation as the operation.
• A covering map p : E → B is a continuous surjection such that each point of B has an open neighbourhood U whose preimage is a disjoint union of open sets, each mapped homeomorphically onto U.

Theorem

• π₁(S¹) ≅ ℤ.
• Brouwer: every continuous map from the closed disc to itself has a fixed point.
• The fundamental group of a wedge of two circles is the free group on two generators.
• For path-connected, locally path-connected and semilocally simply connected spaces, connected covering spaces correspond to conjugacy classes of subgroups of π₁.

Common misconceptions

• Thinking fundamental groups are always abelian.
• Believing spaces with isomorphic fundamental groups are homeomorphic; S² and a point both have trivial fundamental group.

Leads to

Optional discoveries

• Homotopy type theory In homotopy type theory the circle is a higher inductive type, and π₁(S¹) ≅ ℤ has a synthetic proof.
• ← Complex numbers
• ← Functors

Learning pathways

• Graduate preparation
• Geometry and topology

Further reading

• Allen Hatcher, Algebraic Topology (2002)
• James R. Munkres, Topology