We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite dimensional manifold. We succeed for a particular type of piecewise-smooth loops.

We also examine the action of the diffeomorphism group of the circle. It is not a useful action on the manifold that we define. We consider how one might fix this problem and conclude that it can only be done by completing to the space of loops of bounded variation.

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We compare various different definitions of the category of smooth objects. The definitions compared are due to Chen, Frölicher, Sikorski, Smith, and Souriau. The purpose of the comparison is to answer two questions: which, if any, captures the essence of smoothness? and how are the various categories related to each other? The first is a subjective question and we give a subjective answer: Frölicher spaces. For the second we give a more objective answer in terms of the existence of functors between the categories with certain nice properties, such as the existence of inverses and of adjunctions. We also make some general remarks about the role that topology does or doesn't play and on how one might go about defining tangent and cotangential structures on Frölicher spaces.

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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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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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with Sarah Whitehouse

We describe the families of degree zero operations in the first Morava K-theory, and consequently in mod p K-theory, in terms of Adams' operations.

When devising problems for a linear algebra course it is desirable to have an extensive source of integral matrices of determinant one. In this short paper we give a straightforward method of generating some such matrices from binomial coefficients.

We construct a natural inner product on the cotangent spaces of the free loop space of a Riemannian manifold. As an application of, and motivation for, this construction we show how to construct a Dirac operator on the loop space of a string manifold.

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