Algebraic and Geometric Topology

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

DMS-0204169 Laurence R. Taylor William G. Dwyer E. Bruce Williams This is a proposal in algebraic topology, and deals primarily with questions in stable and unstable homotopy theory. The stable side has to do with the study of ring spectra. One goal is to study duality in stable homotopy theory and relate it to classical types of duality for commutative rings; a possible outcome is a clearer understanding of Gross-Hopkins duality. Another goal is to study spaces of associative ring structures from a general deformation theory point of view. On the unstable side, the proposal envisages a conceptual classification of p-compact groups (especially at the prime 2), a comprehensive theory of spaces of natural transformations between various familiar functors, and a study of the homology of homotopy limits. There are also geometric topology components involving smoothings of four-space, L-group calculations, and questions related to Riemann-Roch theorems for Waldhausen K-theory. In a general sense, the proposal deals with the problem of understanding geometric shapes, especially shapes with more than three dimensions. These shapes (called spaces) can arise, for instance, as the collection of all states of a physical system, or the collection of all configurations of a complicated mechanical device. The basic goal is to develop a method for describing spaces in terms of numbers and other algebraic objects, in the expectation that this will make it possible to quantify the subtle geometric characteristics of spaces, and allow for the study of relationships betweens spaces of various types. Some spaces, called manifolds, are particularly symmetrical, and among other things the proposal involves studying the associated symmetry properties of their algebraic invariants.