Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces BSU , BU , BSO, BO, BSp, BGL ∞ (R) + and Q 0 (S 0 ). We show that these infinite loop spaces are the zero spaces of non-unital E ∞ -ring spectra. We introduce the notion of q-nilpotent K-theory of a CW-complex X for any q ≥ 2, which extends the notion of commutative K-theory defined by Adem-Gómez, and show that it is represented by Z × B(q, U ), were B(q, U ) is the q-th term of the aforementioned filtration of BU .For the proof we introduce an alternative way of associating an infinite loop space to a commutative I-monoid and give criteria when it can be identified with the plus construction on the associated limit space. Furthermore, we introduce the notion of a commutative I-rig and show that they give rise to non-unital E ∞ -ring spectra.

This article compares the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM S-modules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω ∞ . We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors Ω ∞ agree. This comparison is then used to show that two different constructions of the spectrum of units gl 1 R of a commutative ring spectrum R agree.

Abstract. For any perfect fibration E −→ B, there is a "free loop transfer map" LB + −→ LE + , defined using topological Hochschild homology. We prove that this transfer is compatible with the Becker-Gottlieb transfer, allowing us to extend a result of Dorabia la and Johnson on the transfer map in Waldhausen's A-theory. In the case where E −→ B is a smooth fiber bundle, we also give a concrete geometric model for the free loop transfer in terms of Pontryagin-Thom collapse maps. We recover the previously known computations of the free loop transfer due to Schlichtkrull, and make a few new computations as well.

In the 50 years since the advent of X-ray astronomy there have been many scientific advances due to the development of new experimental techniques for detecting and characterising X-rays. Observations of X-ray polarisation have, however, not undergone a similar development. This is a shortcoming since a plethora of open questions related to the nature of Xray sources could be resolved through measurements of the linear polarisation of emitted X-rays. The PoGOLite Pathfinder is a balloon-borne hard X-ray

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