|
Abstract:
|
The main objects studied in this doctoral thesis are quasitoric manifolds and
spaces arising as the images of polyhedral product functors. Quasitoric manifolds are
particularly interesting as topological generalization of non-singular toric varieties.
They are a research topic of many mathematical disciplines including toric geometry,
symplectic geometry, toric topology, algebraic geometry, algebraic topology, theory
of convex polytopes, and topological combinatorics. These objects have already
found numerous applications in mathematics and sciences and they continue to be
intensively studied.
In this thesis we put some emphasis on combinatorial methods, focusing on
the interaction of the geometry of toric actions and combinatorics of simple polytopes.
This connection of geometry and combinatorics is based on the fundamental
observation that convex polytopes naturally arise as orbit spaces of toric actions
on quasitoric manifolds. Our main original contributions in this thesis are related
to classical topological questions about degrees of maps between manifolds as well
as their embeddings and immersions into Euclidean spaces. We follow the general
scheme characteristic for Algebraic Topology where a topological problem is reduced,
often by non-trivial reductions, to a question of arithmetical, algebraic, or combinatorial
nature. We believe that the novel applications of this scheme developed in
the thesis, especially the new techniques and calculations, have a potential to be
applied on other problems about quasitoric manifods.
Here is a summary of the content of the thesis. For the reader’s convenience and
for completeness, in the first three chapters we give an elementary exposition of the
basic theory of simplicial complexes, convex polytopes, toric varieties and quasitoric
manifolds. The emphasis is on the fundamental constructions and central results,
however the combinatorial approach, utilized in the thesis, allows us present the
theory in a direct and concrete way, with a minimum of topological prerequisites.
The mapping degrees of maps between quasitoric manifolds are studied in Chapter
4 with a particular emphasis on quasitoric 4-manifolds. Utilizing the technique
pioneered by Haibao Duan and Shicheng Wang, which is based on the intersection
form and the cohomology ring calculations, we demonstrate that a complete information
about mapping degrees can be obtained in many concrete situations. The
theorems and the corresponding criteria for the existence of mapping degrees are
formulated in the language of elementary number theory. It is amusing that the question
whether a number appears as a mapping degree between concrete 4-manifolds is
directly linked with classical results from number theory such as whether a number
can be expressed as a sum of two or three squares, etc. This approach allows us to
analyze many concrete 4-manifolds, including CP2, CP2♯CP2, S2×S2, etc. In Chapter
5 we calculate the Stiefel-Whitney classes of some concrete quasitoric manifolds
and their duals. This information is used to determine cohomological obstructions
to embeddings and immersions of these manifolds in Euclidean spaces. As an initial
observation we showed that the calculations are highly dependent on the action of
torus. Indeed, there are examples of quasitoric manifolds over the same polytope
which exhibit a very different behavior and different complexity of the associated
characteristic classes. Focusing on the quasitoric manifolds over the n-dimensional
cube, we are able to produce quasitoric manifolds which are very complex in the
sense that they almost attain the theoretical minimum dimension for their embedding
or (totally skew) immersion in Euclidean spaces. The thesis ends with an
appendix with an outline of the theory of group actions and equivariant topology. |