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EPSRC Reference: GR/A00355/01
Title: AF: THE TOPOLOGY OF THE STABLE MAPPING CLASS GROUP AND APPLICATIONS IN QUANTUM FIELD THEORY
Principal Investigator: Tillmann, Professor U
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Department: Mathematical Institute
Organisation: University of Oxford
Scheme: Advanced Fellowship (Pre-FEC)
Starts: 01 April 2000 Ends: 30 June 2003 Value (£): 88,181
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Summary on Grant Application Form
Let Fg be a surface of genus g. There are two classical objects associated to Fg which were introduced in the last century. One is the mapping class group Tg, the group of connected components of the orientation preserving diffeomorphisms of Fg. The other is the moduli space Mg of Riemann surfaces of genus g. They are very closely related and I propose to study the associated space BT+oo which, by a theorem of Harer, has through a range of dimensions the same homology as Tg and the same rational homology as Mg. Apart from some results in low dimensions, one only knows a family of torsion classes [CL] and a family of rational classes detected by Miller, Morita and Mumford, in the homology of BT+oo.In my latest work I have shown that BT+oo has an infinite loop space structure and detected another large family of torsion homology classes. Indeed the precise topological statement of this result is much stronger. My intentions for the immediate future are to analyse further the infinite loop space structure. One of the goals is to interpret also the family of rational classes in a topological way and possibly show that these are the only rational homology classes. This would confirm a conjecture by Mumford made some fifteen years ago.From another point of view, as one of my results states, BT+oo is the classifying space of Segal's category of Riemann surfaces [S]. The significance of this and any result that can be obtained by BT+oo for conformal field theory and string theory ought to be explored. One might think of string algebras as the objects studied in Floer homology and quantum homology. In one project I propose that there should be operations on Floer homology induced by the operations on the homology of BT+oo coming from the infinite loop space structure. In a letter to Witten [C], Connes suggests a relation between string theory and Waldhausen K-theory [Wa]. Such a connection would be most interesting but is at this point highly conjectural. Nevertheless, Waldhausen K-theory has come up naturally in my investigation of the topology of BT+oo, and I suspect that this link can be pushed further.
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