Summer Research Fellowship Programme of India's Science Academies

A Literature Review on The Milnor's Book "Topology from the Differentiable Viewpoint"

Gobinda Debnath

BS MS Mathematics, 4th year, 8th semester, NIT Agartala, Barjala Jiraniya, West Tripura, Tripura-799046

Dr Ved Datar

Department of Mathematics, Assistant Professor, Indian Institute of Science (IISc) Bangalore, CV Raman Rd, Bengaluru, Karnataka 560012


In my project I am studying the book "Topology from the differentiable viewpoint" written by Jhon Willard Milnor. He is an american mathamatician known for his work in Differential Topology. He won Fields medal (1962), Wolf prize (1989) and Abel prize (2011). Basically, Differential Topology is the study of those properties of finite dimensional manifolds which are invariant under diffeomorphisms. It deals with differentiable functions on smooth manifolds. Here, the aim was to explore the book chapterwise and study the concepts of tangent space, manifolds, regular values, role of smooth mapping between smooth manifolds (Brouwer Fixed Point Theorem, Sard's Theorem, Brouwer degree of mapping, Degree is Homotopy Invarient), concept of oriented manifolds and its role in giving degree as an integer rather than integer modulo 2.

Keywords: Diffeomorphism, smooth mapping, smooth manifold, regular values, Orientable manifolds, degree of mappings.

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