Classical Homotopy Theory, Simplicial Groups, and Related Structures

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DMS-0305094Frederick R. CohenThe PI intends to study problems with origins in classicalhomotopy theory as well as their overlap with function spaces,cohomology of groups, and properties of classical configuration spaces. The specific problems are as follows:(1) Extend the P.I.'s recent solution of a 25 year oldconjecture of M.G. Barratt on the growth of torsion inthe homotopy groups of certain finite complexes.(2) Analyze these structures for other finite complexesthrough simplicial models for iterated loop spaces asdeveloped by the P.I., and R. Levi.(3) In joint work of the P.I., J. Berrick, Y. Wong, and J. Wu,continue to analyze the interplay between braid groups, andthe homotopy groups of the 2-sphere. These connections extendin joint work with D. Cohen to analogues of classical modularforms obtained from the cohomology of the braid groups withcoefficients in a symplectic representation.(4) In joint work with A. Adem, and D. Cohen, continue toanalyze the structure of spaces of representations offundamental groups of complements of complex hyperplanearrangements, and their associated vector bundles.One of the main directions here is a connection betweenArtin's braid groups, and the classical problem of enumeratingways of continuously wrapping spheres of any dimension aroundother geometric objects. The methods can be thought of asenumerating topological properties of many particles, notallowed to collide, moving through time. The methodsoverlap with several mathematical structures givencombinatorially as well as number theoretically. Thesemethods involve the "geometry of braids", and how thisstructure encodes counting problems through connections toother mathematical objects.