Whitehead theorem

The Whitehead theorem states that a weak homotopy equivalence between CW complexes is a homotopy equivalence.

CW complex: A space built by attaching disks in increasing dimensions, with a topology determined by its cells. The theorem’s hypothesis restricts both spaces to this well-behaved class.
Cellular approximation theorem: A theorem allowing maps between CW complexes to be homotoped to preserve their skeleta. Cellular control makes the map’s extension across successive cells manageable.
J. H. C. Whitehead: A British mathematician whose work helped establish modern homotopy theory and geometric topology. The theorem is named for Whitehead, who proved foundational forms of it.
Classifying space: A space whose bundles or principal bundles encode objects classified by a chosen group or structure. Weak equivalences between CW models preserve the homotopy type used in classification.
Quillen equivalence: An equivalence between homotopy theories presented by model categories. It generalizes the idea of recognizing equivalence from homotopical data across categories.
Weak homotopy equivalence: A map inducing a bijection on path components and isomorphisms on all homotopy groups. This is the weaker condition the theorem upgrades to a homotopy equivalence.
Homotopy extension property: A property allowing a homotopy on a subspace to extend over the ambient space. It lets cell-by-cell homotopies assemble into homotopies on the whole complex.
Homotopy theory: The study of spaces and maps up to continuous deformation. The theorem became a basic tool in this broader mathematical field.
Eilenberg–Mac Lane space: A connected space with one nonzero homotopy group, in a chosen degree. Maps into these spaces are compared through homotopy-group data.
Model category: A category equipped with classes of weak equivalences, fibrations, and cofibrations satisfying compatibility axioms. Model categories formalize settings where Whitehead-type criteria can be applied.

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