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Details of Grant 

EPSRC Reference: GR/S94667/01
Title: The Homotopy theory of fusion systems
Principal Investigator: Levi, Professor R
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematical Sciences
Organisation: University of Aberdeen
Scheme: Standard Research (Pre-FEC)
Starts: 01 October 2004 Ends: 30 September 2007 Value (£): 145,002
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:  
Summary on Grant Application Form
The main theme of this proposal is a new concept in Algebraic Topology -- p-Local Finite Groups. The concept was introduced by the PI and his collaborators C. Broto (Barcelona) and B. Oliver (Pans) in a paper carrying the title of this proposal, which will appear in the October 2003 issue of Journal of the American Mathematical Society. The concept of a p-Local Finite Group is one which links Finite Group Theory to Homotopy Theory and gives a systematic framework where the rich interplay between p-Local Homotopy Theory of classifying spaces and algebraic phenomena in Representation Theory, Group Theory and Group Cohomology can be studied. The new concept has already yielded some beautiful results, but furthermore, it represents a very exciting breakthrough in the subject with vast opportunity for exploration.A p-local finite group consists of a p-group S together with a pair of categories, one which encodes a fusion pattern between subgroups of S, and another which contains just enough extra data to be able to equip this fusion pattern with a classifying space . These classifying spaces are shown in previous work by the PI and his coauthors [BLO2] (see case for support) to share many of the same properties as p-completed classifying spaces of finite groups. In particular any finite group gives rise to a p-local finite group, but there are also many exotic exmaples. In this project we propose further investigation of these objects. We describe briefly here (and in more detail in the Case for Support) two of the central themes.An important outstanding problem in the study of classifying spaces is to provide a manageable algebraic description of mapping spaces between them. in [BLO2] the authors study spaces of equivalences, using what we call a centric linking system , which is the core of the the concept of a p-local finite group. However, to proceed any further, a generalization is required, namely, centric linking systems should be replaced by full linking systems . A topological version of this generalization is already available, but to bring the power of the theory to the full, a discrete construction is required. Once such a construction is achieved, it will have implications to many aspects of the theory, particularly to the study of mapping spaces, but also to a systematic theory of homology decompositions (see Case for Support, for an explanation of this concept).Another crucial aspect of the theory is it's relationship to purely algebraic concepts such as Representation Theory and Group Cohomology. In particular we propose to develop the concepts of a module over a p-local finite group, study the appropriate analogue of group cohomology with coefficients in this module, and relate the results to known facts about classical group cohomology, with the hope of using the added generality to deduce new results on classical questions.Funds are requested mainly to cover the cost of a full time RA for 36 months. This proposal fits well with the Council's policy of encouraging interdisciplinary research (within mathematics).
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Organisation Website: http://www.abdn.ac.uk