The topology of Lie groups

Lead Research Organisation: University of Southampton
Department Name: Sch of Mathematical Sciences

Abstract

Lie groups are fundamental objects in mathematics, with
wide-ranging applications. The topology of Lie groups is concerned with
the properties they have which remain unchanged when the Lie group is
deformed continuously. The study of the topology of Lie groups is a long-running
theme in algebraic topology. This project is concerned with the maps from a
fixed space X into a Lie group G; this itself forms an important topological space
called the mapping space Map(X,G). It is known that if G decomposes as a
product or X decomposes as a wedge then Map(X,G) has a corresponding
decomposition. But little is known about the structural properties of this
decomposition, for example, whether it preserves the multiplication. The
aim of this project is to investigate these structural properties and determine
which properties of X or G are needed to induce them.

Publications

10 25 50

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509747/1 01/10/2016 30/09/2021
1948950 Studentship EP/N509747/1 28/09/2017 30/09/2020 Holly Paveling
 
Description The focus of this award is to identify the set of homotopy classes of H-maps between two Lie groups G and L, when localised at an odd prime p. As yet, we have made the following findings. In the case where G is homotopy commutative, we have (under certain conditions) identified the set of homotopy classes to be a calculable product of homotopy classes of L. We have also found conditions for all maps from G to L to be H-maps up to homotopy. When G is p-regular but not homotopy commutative, we prove analogous results for the classical Lie groups, and provide some examples in the exceptional case.
Exploitation Route These results have largely academic impact; others working with the same classes of groups may be able to utilise properties of H-maps, or use the calculated expressions for the sets of homotopy classes. The results could also possibly be extended to larger classes of groups.
Sectors Other