Nearly trivial homotopy classes between finite complexes
Let \(f:X \rightarrow Y\) be an essential map between finite complexes \(X\) and \(Y\). The authors study a concept which is a version of phantom maps, but for finite complexes. Thus any map has a class \(\mathcal K(f)\) consisting of all spaces \(K\) such that \(f \circ h\) is homotopically trivial for any map \(h: K \rightarrow X\). The authors define two homotopy invariants for \(f\), called the \textit{cone length} and the \textit{killing length} of \(f\) and denoted respectively by \(T_c(f)\) and \(T_k(f)\). (The names are mine, the authors provide the symbols, but not the names for \(T_c(f)\) and \(T_k(f)\), which makes the initial reading of this paper slightly challenging). The map has cone length \(n\) if every space of cone length \(n\) is contained in the class of spaces \(\mathcal K(f)\). Similarly, the map \(f\) has killing length \(n\) if every space of killing length \(n\) is contained in the class of spaces \(\mathcal K(f)\). A map of killing length \(n\) is called a map which is \textit{trivial of order} \(n\) by the authors, but map of cone length \(n\) is not given a name. The cone and killing lengths of spaces have been used by various authors. Consider a cofibration whose cofibre is a wedge of spheres. The smallest sequence of such cofibrations is either the cone length or the killing length of a space \(K\) depending on whether the space is at the end or at the beginning of the sequence, while the contractible space is at the other end. The authors use the concept of a map being trivial of order \(n\) to find a relationship between the skeleta of infinite dimensional domains of essential phantom maps and finite dimensional domains of maps which are trivial of order \(n\). Hence they can show for any \(n\) that there is a map which is trivial of order \(n\) or greater from a quotient of a complex projective space by \(S^2\) to a suitable Moore space.
scientific article; zbMATH DE number 1879120
scientific article