Indefinite String Structure

[摘要] The orthogonal group O(n) appears as the structure group of the frame bundle of an n- dimensional Riemannian manifold. Recently there has been a lot of interest in considering k-connected covers of O(n), or of its stable version O, where the first stages are named Spin(n) for k = 1 and String(n) for k = 3. In this thesis we study the problem in the indefinite case: considering connected covers of the indefinite orthogonal group O(p,q), which appears as structure group of frame bundles of semi-Riemannian manifolds. We thus regard Spin(p, q) and String(p, q) as topological groups up to homotopy equivalence using the Whitehead tower as 1-connected and 3-connected covering with certain conjectures. Then the obstruction for semi-Riemannian manifolds to admit Spin and String groups as their structure groups will be computed in terms of cohomology classes of the corresponding classifying spaces BSpin(p, q) and BString(p,q). While Spin groups are finite dimensional Lie groups, String groups as topological groups are not finite dimensional. We conjecture that we could categorify them to finite dimensional Lie 2-groups, providing some clarifications on their generalizations, namely 2-groupoids, along the way.

[发布机构] the University of Pittsburgh