Statement
More accurately, we suppose given CW complexes X and Y, with respective base points x and y. Given a continuous mapping
such that f(x) = y, we consider for n ≥ 0 the induced homomorphisms
where πn 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 homomorphisms f* are all isomorphisms. The Whitehead theorem then states that a weak homotopy equivalence, for connected CW complexes, is a homotopy equivalence.
Read more about this topic: Whitehead Theorem
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