Algebraic Topology Special Session, BMC 2007

Infinite dimensional Riemannian manifolds have traditionally been divided into two types: strong and weak. One can generalise many of the standard constructions of Riemannian geometry to strong Riemannian manifolds but not to weak ones. In this talk I shall examine some of the basic results of Riemannian geometry in order to refine this classification. The purpose is to show that some constructions can be generalised to certain weak Riemannian manifolds. I shall conclude by describing which level of structure is available for the free loop space of a finite dimensional manifold and by explaining how this can be used in applications.

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Generalised cohomology theories are an important tool in algebraic topology. Part of the study of the theories themselves involves looking at their spaces of "operations" and "co-operations". In a paper in the Algebraic Handbook of Topology, Boardman, Johnson, and Wilson described the former as a comonad in a suitable category and the latter as an "enriched Hopf ring". The former of these is very elegant but not all that intuitive, the latter is less elegant and similarly not intuitive - at least, the "enriched" part. The missing part of these descriptions is a non-linear "tensor product", which was actually introduced by Tall and Wraith in 1970. I shall explain the various descriptions and show how both the unstable operations and co-operations have concise and straightforward descriptions as algebraic objects.

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Based on: Stable and Unstable Operations in mod p Cohomology Theories.

One of the pieces of baggage that comes with a graded cohomology theory is the family of operations. These are self-maps of the cohomology groups obeying certain obvious naturality conditions. There are two main types of operation: stable and unstable. An unstable operation acts only on the cohomology groups of a particular degree whilst a stable operation acts on the cohomology groups of any degree compatibly with the suspension isomorphism. It is clear, therefore, that a stable operation defines a family of unstable ones. However, even if one knows that an unstable operation came from a stable one it may not be easy to reconstruct that stable operation. What is remarkable about the Morava K-theories is that there is a straightforward way to do this.

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This is an introduction to the subject of the differential topology of the space of smooth loops in a finite dimensional manifold. It began as background notes to a series of seminars given at NTNU and subsequently at Sheffield. The topics covered are: the smooth structure of the space of smooth loops; constructions involving vector bundles; submanifolds and tubular neighbourhoods; and a short introduction to the geometry and semi-infinite structure of loop spaces. It is meant to be readable by anyone with a good grounding in finite dimensional differential topology.

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I shall start by explaining just what "semi-infinite" means and how it gives rise to interesting structures in geometry. These structures are studied as part of String Theory and are closely related to Floer Theory. Manifolds which carry a "semi-infinite" structure include loop spaces and the theory seems to have particular simplicities when the original manifold is symplectic.

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In this seminar, we shall introduce semi-infinite manifolds and show how one may define cohomology theories dependent on this semi-infinite structure. In particular, we shall define a de Rham theory. With certain alterations, this is calculable for Wiener manifolds. The goal, however, is to calculate the theory for loop spaces and this is, as yet, an unsolved problem.

These were part of a short series of seminars given by myself and Paul Cooper, another graduate student at Warwick at the time, aiming to explain the connections between Mechanics and Geometry.

The original series was three lectures, of which I gave the first and third. Later, I gave two more seminars on similar themes.

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This seminar was given as a pre-seminar. John Jones was due to give a series of seminars on Elliptic Cohomology in the Geometry Seminar Series at Warwick and so I gave this seminar in the Junior Geometry Series as an introduction aimed at a lower level than John's talks were going to be aimed at.

The plain for this seminar was to go through an example of Elliptic Cohomology in action. The example chosen is called the A-hat-genus. It is actually a degenerate case and so is not strictly an elliptic genus. However, it is still a good introductory example.