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This paper provides certain computations with transfer associated with projective bundles of \mathrm {Spin} vector bundles. One aspect is to revise the proof of the main result of [Trans. Amer. Math. Soc.349 (1997), pp. 4385-4399] which says that all fourfold products of the Ray classes are zero in symplectic cobordism. [ABSTRACT FROM AUTHOR]

For any prime p, the theory of p-local compact groups is modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups and generalises the earlier concept of p-local finite groups. These objects have maximal tori and Weyl groups, although the Weyl groups need not be generated by pseudoreflections. In this paper, we study the rational p-adic cohomology of the classifying space of a p-local compact group and prove that just as for compact Lie groups, it is isomorphic to the ring of invariants of the Weyl group action on the cohomology of the classifying space of the maximal torus. This is applied to show that unstable Adams operations on p-local compact groups are determined in the appropriate sense by the map they induce on rational cohomology. [ABSTRACT FROM AUTHOR]

We give a new construction of a classifying space for the fibre of the double suspension and an elementary proof of Gray's conjecture at odd primes. [ABSTRACT FROM AUTHOR]

Let p: E . B be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut1 (p) denote the identity component of the space of all fibre-homotopy self-equivalences of p. Let Baut1 (p) denote the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut1(p), connecting the results of recent work by the authors and others. We use this model to give classification results for the rational homotopy types represented by Baut11(p) and also to obtain conditions under which the monoid aut11(p) is a double loop-space after rationalization. [ABSTRACT FROM AUTHOR]