Quasitoric manifolds in equivariant complex bordism

Our aim is to study the role of omnioriented quasitoric manifolds in equivariant complex bordism. These are a well-behaved class of even-dimensional smooth closed manifolds with the action of a half-dimensional compact torus and an equivariant stably complex structure. They are beneficial objects to work with as they can be described completely in terms of combinatorial data.We include an overview of equivariant complex bordism, highlighting the relationship between localisation and restriction to fixed point data. By keeping in mind the particularly interesting case when the group in question is the compact torus, we revisit work found in [BPR10], reinterpreting and expanding certain results relating to the universal toric genus.We then consider oriented torus graphs of stably complex torus manifolds and classify these using a boundary operator on exterior polynomials related to geometric equivariant complex bordism classes of the manifolds. We also extend the connected sum construction of quasitoric pairs which allows for a more general notion of the equivariant connected sum of omnioriented quasitoric manifolds.We then consider whether an equivariant version of Buchstaber and Ray?s result in [BR98] holds; that is, does there exist an omnioriented quasitoric manifold in every geometric equivariant complex bordism class in which they naturally exist? We conjecture that this is true showing that we have a combinatorial model for such objects and exhibiting low-dimensional examples.