We introduce open quasitoric manifolds and their functorial properties, including complex bundle maps of their stable tangent bundles, and relate these new spaces to the standard constructions of toric topology: quasitoric manifolds, moment angle manifolds and polyhedral products. We extend the domain of these constructions to countably infinite simplicial complexes, clarifying and generalising constructions of Davis and Januszkiewicz. In particular we describe final spaces in the categories of open quasitoric manifolds and quasitoric spaces, as well as in the categories of characteristic pairs and dicharacteristic pairs. We show how quasitoric manifolds can be constructed smoothly as pullbacks of the final spaces QT(n) for n >= 1, and how stably complex structure also arises this way. We calculate the integral cohomology of quasitoric spaces over Cohen-Macaulay simplicial complexes, including the final spaces QT(n) as a special case. We describe a procedure for calculating the Chern numbers of a quasitoric manifold M and, relating this to our cohomology calculations, show how it may be interpreted in terms of the simplicial homology of H(n), the simplicial complex underlying QT(n).

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19469号 / 理博第4129号 / 新制||理||1594(附属図書館) / 32505 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 岸本 大祐, 教授 加藤 毅, 教授 藤原 耕二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This dissertation demonstrates a procedure to view any quasitoric manifold as a “minimal” sub-manifold of an ambient quasitoric manifold of codimension two via the wedge construction applied to the quotient polytope. These we term wedge quasitoric manifolds. We prove existence utilizing a construction on the quotient polytope and characteristic matrix and demonstrate conditions allowing the base manifold to be viewed as dual to the first Chern class of the wedge manifold. Such dualization allows calculations of KO characteristic classes as in the work of Ochanine and Fast. We also examine the Todd genus as it relates to two types of wedge quasitoric manifolds. Background matter on polytopes and toric topology, as well as spin and complex cobordism are provided.