Home > Whitehead theorem

Stating it more accurately, we suppose give 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_{*: πn(X,x) → πn(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.}