### MA5315 Differential Topology

#### Course Details

**Description:**

To introduce manifolds, differential (smooth) structures on manifolds and classifying smooth manifolds up to the notion of diffeomorphisms

**CourseContent:**

Revision of multivariate calculus , inverse and implicit function theorems, Smooth manifolds, manifolds with boundary, : definitions ,examples, smooth maps, diffeomorphisms, tangent plane, derivative of smooth maps, immersions, submersions, local immersion and submersion theorems, embeddings and submanifolds, regular and critical points of a smooth maps, regular and critical values of smooth maps, Transversality , homotopy and stability of transversality, Sards theorem, Whitney embedding theorem ( We need to define partition of unity and its existence) The degree modulo 2 of a mapping , Oriented manifolds, The Brouwer degree Vector fields and Euler number, Poincare-Hopf index thorem Differential forms, integration on manifolds, Stoke's theorem on manifolds with boundary and introduction to De-Rham cohomolgy

#### Course References:

TextBooks:

Differential topology : Victor Guillemin and Alan Pollack, Publisher: Prentice-Hall 1974

ReferenceBooks:

1) Topology from Differentiable viewpoint : John W Milnor , Princeton University Press, 1997

2) Differential Manifolds : Antoni A Kosinski Dover Publications, Inc, New York 1993

Prerequisite: MA5310, MA5370,MA5380