Shim, Hyung Bo
(2013)
Indefinite String Structure.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
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.
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| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
18 October 2013 |
| Date Type: |
Publication |
| Defense Date: |
23 April 2013 |
| Approval Date: |
18 October 2013 |
| Submission Date: |
12 August 2013 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
85 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
2-category theory, spin groups, string groups, characteristics classes, Whitehead tower, 2-groups, 2-groupoids, cohomology groups of classifying spaces of orthogonal groups |
| Date Deposited: |
18 Oct 2013 17:38 |
| Last Modified: |
15 Nov 2016 14:14 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/19620 |
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