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Geometric and Combinatorial Aspects of 1-Skeleta

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dc.contributor.advisorTom Braden
dc.contributor.advisorEduardo Cattani
dc.contributor.advisorPaul Gunnells
dc.contributor.authorMcDaniel, Chris Ray
dc.contributor.departmentUniversity of Massachusetts Amherst
dc.date2023-09-22T21:28:28.000
dc.date.accessioned2024-04-26T19:47:22Z
dc.date.available2024-04-26T19:47:22Z
dc.date.issued2010-05-01
dc.description.abstractIn 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.
dc.description.degreeDoctor of Philosophy (PhD)
dc.description.departmentMathematics
dc.identifier.doihttps://doi.org/10.7275/1557374
dc.identifier.urihttps://hdl.handle.net/20.500.14394/38687
dc.relation.urlhttps://scholarworks.umass.edu/cgi/viewcontent.cgi?article=1206&context=open_access_dissertations&unstamped=1
dc.source.statuspublished
dc.subject1-Skeleta
dc.subjectEquivariant cohomology
dc.subjectGKM manifolds
dc.subjectMathematics
dc.subjectStatistics and Probability
dc.titleGeometric and Combinatorial Aspects of 1-Skeleta
dc.typedissertation
dc.typearticle
dc.typedissertation
digcom.contributor.authorMcDaniel, Chris Ray
digcom.identifieropen_access_dissertations/250
digcom.identifier.contextkey1557374
digcom.identifier.submissionpathopen_access_dissertations/250
dspace.entity.typePublication
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