Homotopy Theory

Homotopy theory studies spaces up to continuous deformation, generalizing the fundamental group to higher homotopy groups and organizing maps through fibrations, cofibrations, and CW approximation.

Definition

Homotopy theory studies topological spaces and maps up to homotopy — continuous deformation — using the higher homotopy groups (homotopy classes of maps from spheres) and the structures of fibrations and CW complexes that make these invariants tractable.

Scope

This topic defines higher homotopy groups, which are abelian for dimension at least two, and develops the tools that compute and relate them: fibrations and the long exact sequence of a fibration, the Hurewicz theorem connecting homotopy and homology, Whitehead's theorem on weak equivalences of CW complexes, and obstruction theory. It surveys the (largely open) problem of the homotopy groups of spheres, Eilenberg-MacLane spaces representing cohomology, and the model-categorical viewpoint that frames homotopy theory abstractly.

Core questions

• How do higher homotopy groups extend the fundamental group, and why are they abelian above dimension one?
• How does the long exact sequence of a fibration compute homotopy groups from simpler pieces?
• What does the Hurewicz theorem say about the first nonzero homotopy group and its relation to homology?
• Why are the homotopy groups of spheres so difficult, and what structure organizes them?

Key concepts

• Higher homotopy groups and their abelian structure
• Fibrations, cofibrations, and the long exact sequence of a fibration
• Hurewicz theorem and Whitehead's theorem
• Eilenberg-MacLane spaces and representability of cohomology
• CW approximation and obstruction theory

Clinical relevance

Homotopy theory is the abstract backbone of modern topology and supplies the language of stable phenomena, classifying spaces for bundles and gauge theories, and the homotopical methods now used across algebra, algebraic geometry, and mathematical physics.

History

Hurewicz introduced higher homotopy groups in the 1930s; Serre's spectral sequence and the work of Whitehead and others made computation possible, and Quillen's model categories (1967) abstracted homotopy theory into a framework applicable far beyond topology.

Key figures

• Witold Hurewicz
• J. H. C. Whitehead
• Daniel Quillen

Seminal works

• hatcher2002
• bredon1993

Frequently asked questions

Why are higher homotopy groups abelian but the fundamental group need not be?

For dimension at least two there is enough room to commute two spheroids past each other via the Eckmann-Hilton argument, forcing commutativity; in dimension one loops cannot be slid past each other this way.

Are the homotopy groups of spheres known?

Only partially. Despite enormous effort they are computed only in a range of dimensions, and determining them in general remains one of the deepest open problems in topology.