Module I (12 hours)
Manifolds and Smooth Maps: Derivatives and tangents, The inverse function theorem and immersions, submersions, transversability, homotopy and stability, Sard’s theorem and Morse functions, embedding of manifolds into Euclidean spaces, simplices, simplicial complexes, simplicial surfaces, Euler characteristic, Proof of the classification of compact and connected surfaces, smooth surfaces, tangent and normal vectors, First fundamental forms, directional derivatives. Coordinates free, Dirtectional derivatives-coordinates, length and area, isometries.

Module II (6 hours)
Transversality and interactions: Manifolds with boundaries, one-manifolds and some consequences, Transversality and interaction theorem mod 2, winding numbers and the Jordan-Brouwer separation theorem.

Module III (6 hours)
Oriented interaction theorem: Motivation, orientation, oriented interaction number, Lefschetz fixed-point theory, Vector fields and the Poincare-Hopf theorem, The Hopf degree theorem, The Euler characteristic and triangulations,

Module IV (6 hours)
Integration on manifolds: Introduction, exterior algebra, differential forms, interior of manifolds, exterior derivatives, cohomology with forms, Stokes’ theorem, integration and mappings, the Gauss-Bonnet theorem.

To introduce the topological aspects of smooth manifolds, distinct from the differential geometric aspects.

CO1. Students will learn about smooth manifolds and smooth mappings between manifolds.

CO2. They will get familiar with tangent bundles, embeddings, immersions and submersions.

CO3. They will learn about classification of certain types of surfaces.

CO4. Students will learn about differential forms and integration on manifolds.

CO5. They will get acquainted with the extension of classical results such as Stoke's theorem to manifolds.

D. B. Gauld, Differential Topology: An Introduction, Dover Publication

M. W. Hirsch, Differential Topology, Springer-Verlag

G. E. Bredon, Topology and Geometry, Springer-Verlag

A. Mukherjee, Topics on Differential Topology, Hindustan Book Agency