A Survey Of Topological,Algebraic And C ∗-Algebraic K-Theory

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Fathi Baghbadorani, Ali | 2010

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 40557 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Fanai, Hamidreza
Abstract:
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 groupsKi(R). Apart from the different origins of these theories and vast varity of their applications, this thesis is modest in its goals. After a thorough study of toplogicalK−groups we state the Butt periodicity theorem and calculate K−groups of some familiar spaces. In the next step we establish an isomorphism between K0(X) for a compact hausdorff space X and K0(C(X))for the C∗−algebra of complex valued functions on X and the algebraicK0(C(X))for C(X)as a ring.
Keywords:
Vector Bundle ; K-Theory ; Serre-Swan Theorem ; C* Algebra ; Butt Periodicity Theorem ; Projective Module

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