A continuous map \(f\colon X\to Y\) of topological spaces is a weak homotopy equivalence if \(f_*\colon\pi_0(X)\to\pi_0(Y)\) is a bijection and, for every \(x_0\in X\) and every \(n\geq1\), the induced map on homotopy groups \[ f_*\colon\pi_n(X,x_0)\to\pi_n(Y,f(x_0)) \] is an isomorphism [@Hat02].
Let \(f\colon X\to Y\) be a weak homotopy equivalence.
For every \(n\) and every abelian group \(G\), \(f_*\colon H_n(X;G)\to H_n(Y;G)\) and \(f^*\colon H^n(Y;G)\to H^n(X;G)\) are isomorphisms [@Hat02].
(Whitehead) If \(X\) and \(Y\) are CW complexes, then \(f\) is a homotopy equivalence [@Hat02].