Classical and A1-homotopy theory of linear algebraic groups

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This project studies the interface between algebra and topology. We study the homotopy theory, i.e., the properties that do not change even after continuous deformations, of the symmetry groups of algebraic structures. The proposal is in two parts. The first is when the algebraic objects consist of vector spaces over a field k with no additional structure. In this case, the structure groups the general linear groups GLn, which are comprised of nxn invertible matrices. We study the A1-homotopy theory of GLn. A1-homotopy is a powerful way to define a homotopy theory of algebraically-defined objects. In this theory, first established in the late 1990s, one considers those deformations that may be defined by polynomial functions. In classical homotopy theory, much information about a space X is encoded in its homotopy groups: pi_n(X), which record the different homotopy-classes of continuous functions from spheres to X. In A1-homotopy, one may analogously define homotopy groups, but now the sense of homotopy is the A1-homotopy. The ordinary homotopy groups are difficult to calculate in most cases, and the A1-homotopy groups are even more difficult to determine. The A1-homotopy groups of GLn and related spaces encode subtle and mysterious information about the underlying field k, in the guise of the algebraic K-theory of k, and this proposal will calculate these homotopy groups in order to extract and make sense of that information. We will gain insight into the way in which the theory of vector bundles over an algebraic-geometric object X relates to the algebraic K-theory of X. We will also learn more about the homotopy groups of the spheres themselves, since the group GLn is a symmetry group of the A1-homotopy sphere A^n-0. The second part of the proposal examines what happens when the vector space A has an additional structure, such as multiplication. A is then an algebra, a prevalent structure in mathematics. The symmetries G are restricted by the multiplication of A and are harder to understand than in the case where the multiplication was absent. There are particular geometric spaces associated to the data of (G,A): spaces parametrizing r-tuples of elements in A that are sufficient to generate the entire algebraic structure of A. These spaces are little-studied to date, but because they are algebraically defined, we may use algebraic techniques to examine their ordinary homotopy theory, facilitating a number of explicit calculations. In this way, we will cast light on the symmetry group G and on algebraic structures related to A. The project will use homotopy theory to deepen our fundamental knowledge about several different kinds of widely-used algebraic structures: algebras, algebras with involution, vector bundles on algebraic objects, and fields (through the K-theory). It will also tell us more about the topology of maps between spheres, which are the most fundamental topological objects but about which many questions remain unanswered.