The Discrete Hodge Star Operator and PoincarÃ© Duality
Arnold, Rachel Florence
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This dissertation is a uniï¬ cation of an analysis-based approach and the traditional topological-based approach to PoincarÃ© duality. We examine the role of the discrete Hodge star operator in proving and in realizing the PoincarÃ© duality isomorphism (between cohomology and ho- mology in complementary degrees) in a cellular setting without reference to a dual cell complex. More speciï¬ cally, we provide a proof of this version of PoincarÃ© duality over R via the simplicial discrete Hodge star deï¬ ned by Scott Wilson in  without referencing a dual cell complex. We also express the PoincarÃ© 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 Formanâ s complex of nontraditional differential forms, deï¬ ned in , in the uniï¬ cation of analytic and topological perspectives discussed in this dissertation. In particular, we establish a ring isomorphism (on the cohomology level) between Formanâ 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.
- Doctoral Dissertations