In toric topology it is important to have a way of constructing Delzant polytopes, which have canonical combinatorial data. Furthermore we have examples of quasitoricmanifolds with representatives in all even dimensions. These examples give riseto sequences of polytopes with related combinatorial invariants.In this thesis we intend to formalise the concept of a family of polytopes, whichwill behave in a similar way to the quotient spaces of quasi-toric manifolds. We willthen compute certain combinatorial invariants in this context.Recently, polytope theory has developed to include the ring P with homogeneous polynomialinvariants and an important operator d, which takes a polytope to the disjointunion of its facets. We will examine our families against this background and extendthe polynomial invariants and d to entire families. In particular we will introduce amethod to calculate polynomial invariants of families by the use of partial differentialequations.We will also look at some polytopes called nestohedra, which arise from buildingsets. These nestohedra give us a construction of Delzant polytopes. We will showthat it is possible to calculate d for any given nestohedra directly from its buildingset. 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 asa facet of P.We will see that nestohedra naturally form families. We will end this work bycombining the work on nestohedra with the work on families and calculating thecombinatorial invariants of some families of nestohedra.
|Date of Award
|31 Dec 2010
- The University of Manchester
|Nigel Ray (Supervisor) & Victor Buchstaber (Supervisor)