Sharif Digital Repository

In this thesis we study three versions of K−theory. The most well-known vesrsion is topological K−theory, a generalization of Grothendieck works on algebraic varieties to the topological setting by Atyiah and Hirzebruch. Since its birth it has been an indespensible tool in topology,differential geometry and index theory. In the early 1970s C∗−algebraic version of K−theory introduced through associating two abelian groups,K0(A)and K1(A)to a C∗−algebra like A. These functors proved to be a powerful machine, making it possible to calculate the K−theory of a great many C∗−algebras. At last,algebraic K−theory is dealig with linear algebra over a ring R by associating it, a sequence of abelian... 

Given an element "H" in H^3 (X,Z), one can define the twisted K-group K(X,H). These groups have been appeared in physics and in particular in String Theory. As an explicit example, we study the twisted K-group of principle circle bundle E. An important notion related is the T-duality. We introduce (E,H), with E a principle circle bundle and H an element in H^3 (E,Z). Under certain circumstances we call (E,H) and (E,H)T-Dual. We study twisted K-theory of dual pair (E,H) and (E,H)  

Studying interrelation between geometry and topology has always been an intriguing idea for classifying manifolds; characteristic classes, signature, Euler characteristic, index of Dirac operator are classical examples of topological invariants that use additional structure to be constructed. Index Theory is seeking to develop a literature to unite these invariants. A culmination in this theory happens in Atiyah and Singer index theorem, which gives a tool to relate elliptic operators on manifolds to the topology. Using this theorem, the index of Dirac operator on an spin manifold is independent of the geometric structure and so is decided by the topology. Another tool for analyzing elliptic...