Date of Award

5-2010

Document type

dissertation

Access Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

First Advisor

Tom Braden

Second Advisor

Eduardo Cattani

Third Advisor

Paul Gunnells

Subject Categories

Mathematics | Statistics and Probability

Abstract

In this thesis we investigate 1-skeleta and their associated cohomology rings. 1-skeleta arise from the 0- and 1-dimensional orbits of a certain class of manifold admitting a compact torus action and many questions that arise in the theory of 1-skeleta are rooted in the geometry and topology of these manifolds. The three main results of this work are: a lifting result for 1-skeleta (related to extending torus actions on manifolds), a classification result for certain 1-skeleta which have the Morse package (a property of 1-skeleta motivated by Morse theory for manifolds) and two constructions on 1-skeleta which we show preserve the Lefschetz package (a property of 1-skeleta motivated by the hard Lefschetz theorem in algebraic geometry). A corollary of this last result is a conceptual proof (applicable in certain cases) of the fact that the coinvariant ring of a finite reflection group has the strong Lefschetz property.

DOI

https://doi.org/10.7275/1557374

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