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Whitehead’s theorem
Abbreviated statement: if \(X, Y\) are CW complexes, then any map \(f: X \to Y\) is a weak homotopy equivalence if and only if it is a homotopy equivalence.
(Note: \(f\) must induce maps on all homotopy groups simultaneously.)
Full Statement: If \((X, x_0) \xrightarrow{f} (Y, f(x_0))\) such that the induced maps \begin{align*} f_*: \pi_*(X, x_0) \to \pi_*(Y, y_0) \\ [g] \mapsto [f \circ g] \end{align*} are all isomorphisms and \(Y\) is connected, then \(f\) is a homotopy equivalence.
