Models for torsion homotopy types
The authors generalize the closed model category structures on the category \(Top_*\) of pointed topological spaces, considered by \textit{J. I. Extremiana Aldana}, \textit{L. J. Hernández Paricio} and \textit{M. T. Rivas Rodríguez} [Proc. Am. Math. Soc. 124, No. 11, 3545-3553 (1996; Zbl 0866.55008)]. To this end, they choose any space \(M=\Sigma M'\) which is the pointed suspension of a CW-complex \(M'\), and distinguish the following types of maps \(f:X\to Y\) in \(Top_*\) satisfying the 5 Quillen axioms for a closed model category: (1) \(f\) is a weak \(M\)-equivalence if \(f_*:[\Sigma^rM,X]\to[\Sigma^rM,Y]\) are isomorphisms for \(r\geq 0\); (2) \(f\) is an \(M\)-fibration if \(f\) has the right lifting property with respect to the inclusions \((C\Sigma^rM'\rtimes\{0\})\cup(\Sigma^rM'\rtimes I)\to C\Sigma^rM'\rtimes I\) for \(r\geq 0\) \((X\rtimes I\) essentially denotes the ordinary pointed cylinder); (3) \(f\) is an \(M\)-cofibration if \(f\) has the left lifting property with respect to every trivial fibration. In this closed model category, each map \(*\to X\) has a factoring \(*\to X^M\to X\) where the first map is an \(M\)-cofibration and the second map is a trivial \(M\)-fibration. This yields a colocalization functor, which is a model for an \(M\)-CW-approximation in the sense of [\textit{E. Dror Farjoun}, Cellular Spaces, null spaces and homotopy localization, Lect. Notes Math. 1622 (1995; Zbl 0842.55001)]. The authors investigate this construction and its properties in the special case where, given any set \(P\) of positive integers and a fixed integer \(n\geq 2\), \(M\) is the wedge of torsion Moore spaces of type \(M(\mathbb{Z}/m,n)\) for \(m\in P\). In this case, the space \(X^M\), which now is denoted by \(X^{(P,n)}\), is built from torsion Moore spaces of type \(M(\mathbb{Z}/m,r)\) with \(r\geq n\) and \(m\in P\). Each map \(X\to*\) has a factoring \(X\to X_{(P,n)}\to *\) where the first map is an \((P,n)\)-cofibration and the second map is a trivial \((P,n)\)-fibration. This yields a localization functor assigning to each \(X\) a space \(X_{(P,n)}\) such that \(\pi_r(X_{(P,n)})\) is uniquely \(P\)-divisible for \(r\geq n+1\) and \(P\)-torsion-free for \(r=n\). In a last section the authors prove the homotopy groups of \(X^{(P,n)}\) coincide with those of the homotopy fibre of the localization map \(X\to X_{(P,n)}\) in all dimensions \(r\neq n\). Using this result, they compute \(K(A,d)^{(P,n)}\) for any abelian group \(A\), \(d\geq 1\), and \(P=\{p^k\}\) or \(P=\{p,p^2,p^3,\dots\}\).
scientific article; zbMATH DE number 1239669
scientific article