Nearly trivial homotopy classes between finite complexes

We construct examples of essential maps of finite complexes f :X→Y which are trivial of order ⩾n. This latter condition implies that for any space K with cone length ⩽n, the induced map f∗=0 :[K,X]→[K,Y]. The main result establishes a connection between the skeleta of the infinite dimensional domains of essential phantom maps and the finite dimensional domains of maps which are trivial of order ⩾n. In particular, there are essential maps f :Σ2i(CPt/S2)→M(Z/ps,2l+3) which are trivial of order ⩾n.