Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories

The realizability of operations on homotopy groups concentrated in two degrees

The homotopy groups of a space are endowed with homotopy operations which define the \Pi-algebra of the space. An Eilenberg-MacLane space is the realization of a \Pi-algebra concentrated in one degree. In this paper, we provide necessary and sufficient conditions for the realizability of a \Pi-algebra concentrated in two degrees. We then specialize to the stable case, and list infinite families of such \Pi-algebras that are not realizable.Comment: Version 2: Some minor corrections. A few changes to the exposition. To appear in the Journal of Homotopy and Related Structure

Rigidity and exotic models for v1-local G-equivariant stable homotopy theory

We prove that the v1-local G-equivariant stable homotopy category for G a finite group has a unique G-equivariant model at p=2. This means that at the prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p≥5 equivariant rigidity fails in general

The Morava K-theory Hopf Ring for BP

Let K be a p-local complex-oriented homology theory. The K-homology of the even spaces in the Ω-spectrum for BP form a Hopf ring. In [6] Ravenel and Wilson chararacterise this Hopf ring by a purely algebraic universal property, and also prove that the K-homology of each component of each even space is polynomial under the star product. The star-indecomposables in this Hopf ring form an algebra under th