| Type of Document |
Dissertation |
| Author |
Arnold, Rachel Florence
|
| Author's Email Address |
rlongley@vt.edu |
| URN |
etd-05012012-164117 |
| Title |
The Discrete Hodge Star Operator and Poincaré Duality |
| Degree |
PhD |
| Department |
Mathematics |
| Advisory Committee |
| Advisor Name |
Title |
| Haskell, Peter E. |
Committee Chair |
| Floyd, William J. |
Committee Member |
| Rossi, John F. |
Committee Member |
| Thomson, James E. |
Committee Member |
|
| Keywords |
- Cell Complex
- Cubical Whitney Forms
- Poincaré Duality
- Discrete Hodge Star
|
| Date of Defense |
2012-05-01 |
| Availability |
unrestricted |
Abstract
This dissertation is a unification of an analysis-based approach and the traditional topological-based approach to Poincaré duality. We examine the role of the discrete Hodge star operator in proving and in realizing the Poincaré duality isomorphism (between cohomology and ho-
mology in complementary degrees) in a cellular setting without reference to a dual cell complex. More specifically, we provide a proof of this version of Poincaré duality over R via the simplicial discrete Hodge star defined by Scott Wilson in [19] without referencing a dual cell complex. We also express the Poincaré duality isomorphism over both R and Z in terms of this discrete operator. Much of this work is dedicated to extending these results to a cubical setting, via the introduction of a cubical version of Whitney forms. A cubical setting provides a place for Robin Forman’s complex of nontraditional differential forms, defined in [7], in the unification of analytic and topological perspectives discussed in this dissertation. In particular, we establish a ring isomorphism (on the cohomology level) between Forman’s complex of differential forms with his exterior derivative and product and a complex of cubical cochains with the discrete coboundary operator and the standard cubical cup product.
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| Files |
| Filename |
Size |
Approximate Download Time
(Hours:Minutes:Seconds)
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| 28.8 Modem |
56K Modem |
ISDN (64 Kb) |
ISDN (128 Kb) |
Higher-speed Access |
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Arnold_RF_D_2012.pdf |
697.39 Kb |
00:03:13 |
00:01:39 |
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