Operads, Group Actions, and Classifying Spaces

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DMS-0206963Clarence W. WilkersonJames E. McClureJeffrey H. SmithWilkerson (in joint work with W. G. Dwyer of Notre Dame) studies the classifying spaces of Lie groups and p-compact groups with the goal of finishing the classification of 2-compact groups and their automorphisms. Wilkerson and Smith study actions of finite groups on arbitrary finite complexes. The goal is to construct a moduli spacethat classifies actions with given fixed point data.McClure and Smith will continue their work on chain operads and chain models for homotopy theories. Specifically, they propose to find a small chain model for the framed little-disks operad, to show that the category of "unstable" coalgebras over a certain chain operad is a model for HZ-local homotopy theory of spaces, to give a similar model for HZ-local spectra, and to create a chain model for the model category of K(n)-module spectra. They also propose to investigate the properties of a symmetric monoidalstructure on the category of cosimplicial chain complexes.McClure uses the joint work with Smith to study the homotopytheoretic properties of the Snaith splitting. He hopes to find a simplified proof of the theorem of Goerss-Hopkins theorem which gives the spectrum E(n) a commutative multiplication. He also studies the rational homotopy theory of equivariant spectra when the group is the circle.Smith and Grodal are studying homotopy G-spheres. That is, spaces that are homotopy equivalent to a sphere and have an action of a finite group G. They hope to give a complete classification based on algebraic invariants of the group. They also study the moduli space of homotopy G-spheres.Homotopy theory is the most fundamental of all geometries. It studies those geometric properties which do not change no matter what continuous deformations are made. The "equality"of donuts and coffee cups is a well known example. Yet, surprisingly, geometry as studied by homotopy theory has an intrinsic algebraic nature. The PIs study the geometric properties of spaces using techniques that come from algebra, with homotopy theory providing the bridge between these different areas of mathematics. In fact, all homotopy information of a space can be described using algebra. The algebra is complicated but homotopy theory gives a correspondence between geometry and algebra that has many important applications.