weak homotopy equivalence

A map \(f: X \to Y\) is called a weak homotopy equivalence if the induced maps \(f_i^* : \pi_i(X, x_0) \to \pi_i(Y, f(x_0))\) are isomorphisms for every \(i \geq 0\).

This is a strictly weaker notion than homotopy equivalence - for example, let \(L\) be the long line. Then \(\pi_i(L) = 0\) for all \(i\), but \(L\) is not contractible, and thus \(L \not\sim {\operatorname{pt}}\). However, the inclusion \({\operatorname{pt}}\hookrightarrow L\) is a weak homotopy equivalence, which can not be a homotopy equivalence.

Any weak homotopy equivalence induces isomorphisms on all integral co/homology groups, and thus co/homology groups with any coefficients by the UCT.