Joint with Sarah Whitehouse

We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E^*. Our description is as a graded and completed version of a Tall-Wraith monoid. The E^*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.

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Formerly, Constructing Smooth Loop Spaces

We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions for the model space which, if satisfied, mean that a smooth structure exists.

We also show how various desired properties can be derived from the model space; for example, topological properties such as paracompactness. We pay particular attention to the fact that the loop spaces that can be defined in this way are all homotopy equivalent; and also to the action of the circle by rigid rotations.

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Joint with Sarah Whitehouse

We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. In the main example, where the target theory is one of the Morava K-theories, this provides a simple and explicit description of a splitting arising from the Bousfield-Kuhn functor.

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In this paper we define and examine the truncated Witten genus. It is defined as the equivariant index of the Dirac operator on the manifold Map(C_p,M) with its natural C_p-action. Here, Map(C_p,M) is the space of maps from the cyclic group of order p into a closed, connected, spin manifold, M. By applying the Atiyah-Singer index theorem we give a topological formula for the truncated Witten genus which is related to the formula for the Witten genus by truncation of the infinite products. We also show that the equivariant index of the Dirac operator on the projective space ℙMap(C_p,ℂⁿ⁺¹) is closely related to the truncated Witten genus of ℂℙⁿ. The spaces ℙMap(C_p,ℂⁿ⁺¹) define a filtration of the space ℙMap(S¹,ℂⁿ⁺¹) which has been used to study equivariant objects on the smooth loop space of ℂℙⁿ.

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In this paper we consider vector bundles with fibre a loop space and structure group a loop group. We determine necessary and sufficient conditions for the structure group to reduce to the group of constant loops in terms of the existence of certain finite dimensional subbundles of the original vector bundle.

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with Ralph L. Cohen

In this paper we investigate bundles whose structure group is the loop group LU(n). These bundles are classified by maps to the loop space of the classifying space, LBU(n). Our main result is to give a necessary and sufficient criterion for there to exist a Fourier type decomposition of such a bundle ξ. This is essentially a decomposition of ξ as ζ ⊗Lℂ, where ζ is a finite dimensional subbundle of ξ and Lℂ is the space of functions, C^∞ (S¹,ℂ). The criterion is a reduction of the structure group to the finite rank unitary group U(n) viewed as the subgroup of LU(n) consisting of constant loops. Next we study the case where one starts with an n dimensional bundle ζ→Mclassified by a map f: M → BU(n) from which one constructs a loop bundle Lζ→LM classified by Lf: LM →LBU(n). The tangent bundle of LM is such a bundle. We then show how to twist such a bundle by elements of the automorphism group of the pull back of ζ over LM via the map LM →M that evaluates a loop at a basepoint. Given a connection on ζ, we view the associated parallel transport operator as an element of this gauge group and show that twisting the loop bundle by such an operator satisfies the criterion and admits a Fourier decomposition.

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