Whitehead theorem

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In mathematics, the Whitehead theorem in homotopy theory states that if a continuous mapping f between topological spaces X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence provided X and Y are connected CW complexes. This result was proved by J. H. C. Whitehead, and provides a justification for working with the CW complex concept that he introduced.

Stating it more accurately, we suppose given CW complexes X and Y, with respective base points x and y. Given a continuous mapping

f : X → Y,

such that f(x) = y, we consider for n ≥ 0 the induced mappings

f_* : π_n(X,x) → π_n(Y,y)

where π_n denotes for n ≥ 1 the n-th homotopy group. For n = 0 this means the mapping of the path-connected components; if we assume both X and Y are connected we can ignore this as containing no information. We say that f is a weak homotopy equivalence if the mappings f_* are all bijective. The Whitehead theorem then states that a weak homotopy equivalence, for connected CW complexes, is an actual homotopy equivalence.