In toric topology it is important to have a way of constructing Delzant polytopes, which have canonical combinatorial data. Furthermore we have examples of quasitoric manifolds with representatives in all even dimensions. These examples give rise to sequences of polytopes with related combinatorial invariants. In this thesis we intend to formalise the concept of a family of polytopes, which will behave in a similar way to the quotient spaces of quasi-toric manifolds. We will then compute certain combinatorial invariants in this context. Recently, polytope theory has developed to include the ring P with homogeneous polynomial invariants and an important operator d, which takes a polytope to the disjoint union of its facets. We will examine our families against this background and extend the polynomial invariants and d to entire families. In particular we will introduce a method to calculate polynomial invariants of families by the use of partial differential equations. We will also look at some polytopes called nestohedra, which arise from building sets. These nestohedra give us a construction of Delzant polytopes. We will show that it is possible to calculate d for any given nestohedra directly from its building set. We will also show that the canonical characteristic function of a nestohedron, F, which is a facet of a nestohedron, P, agrees with the characteristic function of F as a facet of P. We will see that nestohedra naturally form families. We will end this work by combining the work on nestohedra with the work on families and calculating the combinatorial invariants of some families of nestohedra.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:525980 |
| Date | January 2010 |
| Creators | Fenn, Andrew Graham |
| Contributors | Ray, Nige ; Buchstaber, Victor |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.research.manchester.ac.uk/portal/en/theses/on-families-of-nestohedra(b3697191-05bc-415f-b66d-b0a945b04927).html |
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