On weak homotopy equivalences between mapping spaces (Q1382165)

Suppose that a map \(h:X\to Y\) induces bijections \([S^n_+, X]\cong [S^n_+, Y]\) for all \(n\geq 0\), where \(S^n_+\) is the \(n\)-sphere with a disjoint basepoint. The authors prove that \(h\) is a weak homotopy equivalence if and only if the induced homomorphism of fundamental groups is bijective on each path connected component. They show that this condition holds in certain situations.