Comparing homotopy categories

Abstract

Abstract. Given a suitable functor T : C → D between model categories, we define a long exact sequence relating the homotopy groups of any X ∈ C with those of T X, and use this to describe an obstruction theory for lifting an object G ∈ D to C. Examples include finding spaces with given homology or homotopy groups. IntroductionA number of fundamental problems in algebraic topology can be described as measuring the extent to which a given functor T : C → D between model categories induces an equivalence of homotopy categories: more specifically, which objects (or maps) from D are in the image of T , and in how many different ways. For example: a) How does one distinguish between different topological spaces with the same homology groups, or with chain-homotopy equivalent chain complexes? How can one realize a given map of chain complexes up to homotopy? b) When do two simply-connected topological spaces have the same rational homotopy type? c) When is a given topological space a suspension, up to homotopy? Dually, how many distinct loop space structures, if any, can a given topological space carry? d) Is a given Π-algebra (that is, a graded group with an action of the primary homotopy operations) realizable as the homotopy groups of a topological space, and if so, in how many ways?Our goal is to describe a unified approach to such problems that works for functors between spherical model categories, for which several familiar concepts and constructions are available. These include a set A of models (to play the role of spheres, in particular determining the corresponding homotopy groups π

Higher homotopy operations and André–Quillen cohomology

“…The map (j n+1 ) # is surjective by the commutative diagram before [Bl4,§4.12], which also shows that there is a short exact sequence:…”

“…Since we assumed thatd n+2 [Bl4,Fact 3.3]). We have thus described a representing map V n+2 → Z n V n+1 • , as required.…”

Comparing cohomology obstructions

“…Remark 8.16. The tower (P n M A X ) ∞ n=0 may be the best approximation to an A-Postnikov tower available, since the category C itself may not have such towers -e.g., when C = T * and A consist of mod-p Moore spaces (see [15,Section 3.10]). …”

On realizing diagrams of Π–algebras

“…However, the description there was in terms of moduli spaces, and it seems worthwhile making obstruction theory explicit. A further generalization of this theory appears in [4], but it is not easy to extract from it the simpler version needed here.…”

“…In particular, if we replace the spheres by Moore spaces as our models (in T * ), then we have neither Eilenberg-Mac Lane objects nor Postnikov systems for the mod p homotopy groups (see Blanc [4,Section 3.10]). In addition, the realization of simplicial spaces does not provide the expected functor J for Ax 4, since the Bousfield-Friedlander spectral sequence for a mod p resolution does not collapse (see Blanc [7,Section 4.6]).…”