On the Complex Cobordism of Flag Varieties Associated to Loop Groups

[摘要] This work is about the algebraic topology of LG/T in particular, the complex cobordism of LG/T where G is a compact semi-simple Lie group. The loop group LG is the group of smooth parametrized loops in G, i.e. the group of smooth maps from the circle S1 into G. Its multiplication is pointwise multiplication of loops. Loop groups turn out to behave like compact Lie groups to a quite remarkable extent. They have Lie algebras which are related to affine Ka?-Moody algebras. The details can be found in [98] and [64]. The class of cohomology theories which we study here are the complex orientable theories. These are theories with a reasonable theory of characteristic classes for complex vector bundles. Complex cobordism is the universal complex orientable theory. This theory has two descriptions. These are homotopy theoretic and geometric. The geometric description only holds for smooth manifolds. Some comments about the structure of this thesis are in order. It is written for a reader with a first course in algebraic topology and some understanding of the structure of compact semi-simple Lie groups and their representations, plus some Hilbert space theory and some mathematical maturity. Some good general references are Kac [60] for Kac-Moody algebra theory, Pressley-Segal [84] for loop groups and their representations, Young [96] for Hilbert space theory, Adams [3] for complex orientable theories, Husemoller [56] and Switzer [93] for fiber bundle theory and topology, Uavenel [87] for Morava K-theories, Lang [74] for the differential topology of infinite dimensional manifolds, Conway [27] for Fredholm operator theory. The organization of this thesis is as follows. Chapter 1 includes all details about Schubert calculus and cohomology of the flag space G/B for Kac-Moody group G. We examine the finite type flag space in section 1. In the section 2, we give some facts and results about Kac-Moody Lie algebras and associated groups and the construction of dual Schubert cocycles on the flag spaces by using the relative Lie algebra cohomology tools. The rest of chapter includes cup product formulas and facts about nil-Hecke rings. Chapter 2 includes the general theory of loop groups. Stratifications and a cell decomposition of Grassmann manifolds and the homogeneous spaces of loop groups are given. In chapter 3, we discuss the calculation of cohomology rings of LG/T. First we describe the root system and Weyl group of LG, then we give some homotopy equivalences between loop grou.ps and homogeneous spaces, and investigate the cohomology ring structures of LSU2/T and ?SU2. Also we prove that BGG-type operators correspond to partial derivation operators on the divided power algebras. In chapter 4, we investigate the topological construction of BGG-type operators, giving details about complex orientable theories, Becker-Gottlieb transfer and a formula of Brumfiel-Madsen. In chapter 5, we develop a version of Quillen's geometric cobordism theory for infinite dimensional separable Hilbert manifolds. For a separable Hilbert manifold X, we prove that this cobordism theory has a graded-group structure under the topological union operation and this theory has push-forward maps. In section 2 of this chapter, we discuss transversal approximations and products, and the contravariant property of this cobordism theory. In section 3, we discuss transversality for finite dimensional fiber bundle. In section 4, we define the Euler class of a finite dimensional complex vector bundle in this cobordism theory and we generalize Bressler-Evens's work on LG/T. In section 6, we prove that strata given in chapter 2 are cobordism classes of infinite dimensional homogeneous spaces. In section 7, we give some examples showing that in certain cases our infinite dimensional theory maps surjectively to complex cobordism.

[发布机构] University:University of Glasgow

[关键词] Mathematics