Certain problems regarding the algebraic topology of manifolds

Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute

Abstract

It often occurs in mathematics that one can get a lot of information about a mathematical object with some structure by considering the space of invertible structure preserving maps from this object to itself. Such spaces, called automorphism groups, are usually equipped with rich structure that makes them interesting objects of study. This approach is particularly fruitful and quite ubiquitous in algebraic topology. The main purpose of the subject, broadly speaking, is to study spaces by means of algebraic invariants. Ideally we wish to work with invariants that are interesting enough to contain a lot of information and useful enough to be easily computable. The rich structure of automorphism groups is, in many cases, helpful in allowing us to reach this balance.

For example, we can consider smooth manifolds. The invertible maps which preserve the smooth structure are the diffeomorphisms and for a smooth manifold M we can define the diffeomorphism group of M, denoted by Diff(M), which is the space of all diffeomorphisms from M to itself. This is an instance of an automorphism group. Apart from the group structure that is shared by all automorphism groups and given by composition and inversion of diffeomorphisms, the diffeomorphism group also has the structure of a topological space. This is a consequence of a more general fact: Given two smooth manifolds M, N,the space of smooth functions from M to N denoted by C sigma(M, N)is a topological space, equipped with the Whitney topology. As a subspace of C sigma(M, N), Diff(M)is a topological space, and furthermore we can show that it is a topological group, meaning that the group operations of composition and inversion are continuous.

The study of diffeomorphism groups of manifolds is an important special case of the study of embedding spaces of manifolds. A notion which arises naturally when thinking about questions regarding embeddings is that of configuration spaces. The configuration space of n points in a manifold M, denoted by Confn(M), is the space of all n-tuples of distinct points of M, or equivalently the space of all possible positions of n particles moving in without colliding. These spaces provide a natural context for talking about embeddings since any embedding F: M-N induces a map on Cartesian products which in turn induces a map on configuration spaces f: Confn(M)-Confn(N). It is reasonable, therefore, to expect that configuration spaces play an important role in constructing invariants of embedding spaces. Indeed, variations of a construction on configuration spaces, the configuration space integral, has given rise to some important invariants of spaces related to embeddings that produced remarkable results in low dimensional and geometric topology.

One of my goals for my graduate studies is to study embedding spaces of manifolds broadly and more specifically understand these constructions better. Despite the important role configuration space integrals have played in the various contexts they have been introduced, the connections between the different known constructions are not yet very well understood. In particular, most known constructions involve integration along fibers of certain differential forms and a general, more homotopy-theoretic, description is lacking. Such a description would be important in allowing us to compare the various constructions of configuration space integrals and would also help us examine related constructions over various coefficients groups. Steps towards this direction, at least for the case of knots, have been taken by Koytcheff [4]. Moreover, we have seen above that the configuration space integrals depend on certain framing data. It would further be interesting to explore what the role of this extra data is and whether we can have similar constructions for other tangential structures [5].

This project falls within the EPSRC Geometry and Topology research area.

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