#### Date of Submission

1-28-2011

#### Date of Award

1-28-2012

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Poddar, Mainak

#### Abstract (Summary of the Work)

The main goal of this thesis is to study the topology of torus actions on manifolds and orbifolds. In algebraic geometry actions of the torus (C * ) n on algebraic varieties with nice properties produce bridges between geometry and combinatorics see [Dan78], [Oda88] and [Ful93]. We see a similar bridge called moment map for Hamiltonian action of compact torus on symplectic manifolds see [Aud91] and [Gui94]. In particular whenever the manifold is compact the image of moment map is a simple polytope, the orbit space of the action. A topological counterpart called quasitoric manifolds, a class of topological manifolds with compact torus action having combinatorial orbit space, were introduced by Davis and Januskiewicz in [DJ91]. A class of examples of quasitoric manifolds are nonsingular projective toric varieties, introduced by M. Demazure [Dem70]. There are many properties of quasitoric manifolds akin to that of nonsingular complete toric varieties. The combinatorial formula for the cohomology ring of a nonsingular complete toric variety is analogous to the formula for quasitoric manifolds. Their K-theories described by P. Sankaran and V. Uma in [SU03] and [SU07] are also similar. The survey [BP02] is a good reference for many interesting developments and applications of quasitoric manifolds.Inspired by the work of [DJ91] we generalize these quasitoric manifolds to quasitoric orbifolds with compact torus action. We have studied structures and topological invariants of quasitoric orbifolds. In addition, we have introduced a class of n-dimensional orbifolds with Z nâˆ’1 2 -action with nice combinatorial description. We have also given two applications of quasitoric manifolds to cobordism theory. This section briefly introduces the main ingredients of this thesis. We will meet all in much greater detail in the following chapters.We recall the definitions and topological invariants namely homology groups, cohomology rings and Chern classes of quasitoric manifolds in Chapter 1. A quasitoric manifold M2n is an even dimensional smooth manifold with a locally standard action of the compact torus T n = U(1)n such that the orbit space has the structure of an n-dimensional simple polytope. Cohomology rings of these manifolds can be computed using equivariant cohomology [DJ91]. As a special case, one obtain the cohomology rings of nonsingular projective toric varieties without recourse to algebraic geometry. Buchstaber and Ray [BR01] showed the existence of smooth and stable almost complex structure on quasitoric manifolds. We present a different proof of this following [Poddf].In Chapter 2 we recall the definitions of small covers and orbifolds. The category of small covers was introduced by Davis and Januszkiewicz [DJ91]. Following them we discuss some basic theory about small covers. The remaining sections of this chapter describe the definition, tangent bundle and orbifold fundamental group of orbifolds following [ALR07]. Orbifolds were introduced by Satake [Sat57], who called them V - manifolds. Precisely, orbifolds are singular spaces that locally look like the quotient of an open subset of Euclidean space by an action of a finite group.In Chapter 3 we study topological invariants and stable almost complex structure on quasitoric orbifolds.

#### Control Number

ISILib-TH308

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Sarkar, Soumen Dr., "Some Aspects of Toric Topology." (2012). *Doctoral Theses*. 66.

https://digitalcommons.isical.ac.in/doctoral-theses/66

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842842