Homotopy Categories for Simply Connected Torsion Spaces

Abstract

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups $$\mathbb{Z} \mathord{\left/ {\vphantom {\mathbb{Z} s}} \right. \kern-\nulldelimiterspace} s$$ with s ∈ S, or in the abelian group $$\mathbb{C}{\left[ {S^{{ - 1}} } \right]} = {\mathbb{Z}{\left[ {S^{{ - 1}} } \right]}} \mathord{\left/ {\vphantom {{\mathbb{Z}{\left[ {S^{{ - 1}} } \right]}} \mathbb{Z}}} \right. \kern-\nulldelimiterspace} \mathbb{Z}$$ where $$\mathbb{Z}{\left[ {S^{{ - 1}} } \right]}$$ is the group of fractions of the form $$\frac{z}{s}$$ with s ∈ S. In the first case, for n > 1 the localized category $${\user2{Ho}}{\left( {\mathcal{T}_{n} S - {\user2{Top}}*} \right)}$$ is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category $${\user2{Ho}}{\left( {\mathcal{T}_{{\mathcal{D}_{n} }} S - {\user2{Top}}*} \right)}$$ is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations $$X^{{\mathcal{T}_{n} S}} \to X$$ , $$X^{{\mathcal{T}_{{\mathcal{D}_{n} }} S}} \to X$$ obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For instance, the map $$X^{{\mathcal{T}_{{\mathcal{D}_{n} }} S}} \to X$$ is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces $$X^{{\mathcal{T}_{n} S}} $$ , $$X^{{\mathcal{T}_{{\mathcal{D}_{n} }} S}} $$ are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If S is generated by a set P of primes and S p is generated by a prime p ∈ P one has that for n > 1 the category $${\user2{Ho}}{\left( {\mathcal{T}_{n} S - {\user2{Top}}*} \right)}$$ is equivalent to the product category $$\Pi _{{p \in P}} {\user2{Ho}}{\left( {\mathcal{T}_{n} S^{p} - {\user2{Top}}*} \right)}$$ . If the multiplicative system S is generated by a finite set of primes, then localized category $${\user2{Ho}}{\left( {\mathcal{T}_{{\mathcal{D}_{n} }} S - {\user2{Top}}*} \right)}$$ is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for $${\user2{Ho}}{\left( {\mathcal{T}_{n} S - {\user2{Top}}*} \right)}$$ .