The Hopf Fibration

Jana, Santanil (2018) The Hopf Fibration. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Santanil Jana (13MS062)) 13MS062.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In this thesis, we will give a brief introduction to homotopy groups and fibrations and discuss the influence of the Hopf map in the development of higher homotopy theory. We will also introduce the notion of Hopf invariant to get a better understanding of the elements of the third homotopy group of the 2- sphere. A detailed discussion about the geometry of the Hopf fibration is also presented. In the course of studying Hopf fibration, we will discuss many important concepts of Algebraic and Differential Topology, namely the Cup products, Orientability, de Rham Cohomology, the degree of smooth maps and so on. Finally, we will establish the notion of linking number in terms of degree and illustrate the fact that the fibers of the Hopf fibration are linked once. We will calculate their linking number rigorously as well as try to give a pictorial argument in favor our calculated result.

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