Homotopy theory

Builds on

Required prerequisites, all the way down: 44

You will learn to

• Define higher homotopy groups, prove they are abelian for n ≥ 2, and compute them from fibrations using the long exact sequence.
• Use the Hopf fibration S¹ → S³ → S² to show π₃(S²) ≅ ℤ.
• State Whitehead's theorem, the Hurewicz theorem and the Freudenthal suspension theorem, and apply them to spheres and CW complexes.

Definition

• πₙ(X, x₀) is the set of homotopy classes of based maps (Sⁿ, *) → (X, x₀), a group for n ≥ 1.
• A map p : E → B is a (Hurewicz) fibration if it has the homotopy lifting property for all spaces.
• A CW complex is a space built inductively by attaching cells Dⁿ along their boundary spheres.
• An Eilenberg–MacLane space K(G, n) is a connected CW complex whose only non-trivial homotopy group is πₙ ≅ G.

Theorem

• A fibration F → E → B gives a long exact sequence ⋯ → πₙ(F) → πₙ(E) → πₙ(B) → πₙ₋₁(F) → ⋯.
• πₖ(Sⁿ) = 0 for k < n and πₙ(Sⁿ) ≅ ℤ; π₃(S²) ≅ ℤ, generated by the Hopf map.
• Whitehead: a weak homotopy equivalence between CW complexes is a homotopy equivalence.
• The homotopy groups of spheres πₙ₊ₖ(Sⁿ) are known only in a limited range; their complete computation is a major open problem.

Common misconceptions

• Thinking higher homotopy groups of spheres vanish above the dimension as homology does; π₃(S²) ≅ ℤ.
• Believing that a map inducing isomorphisms on all homology groups is a homotopy equivalence; this needs extra hypotheses such as simple connectivity of CW complexes.

Leads to

Optional discoveries

• ∞-categories ∞-categories are a framework for doing homotopy theory systematically: spaces themselves are the ∞-groupoids.
• Homotopy type theory Homotopy type theory proves results such as π₁(S¹) ≅ ℤ and the Freudenthal suspension theorem synthetically, inside type theory.

Learning pathways

• Geometry and topology

Further reading

• Allen Hatcher, Algebraic Topology (2002)
• J. P. May, A Concise Course in Algebraic Topology (1999)