Course Details
Subject {L-T-P / C} : MA5112 : Differential Topology { 3-0-0 / 3}
Subject Nature : Theory
Coordinator : Prof. Divya Singh
Syllabus
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. 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. 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, 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.
Course Objectives
- To introduce the topological aspects of smooth manifolds, distinct from the differential geometric aspects.
Course Outcomes
Students will learn about smooth manifolds, smooth mappings between manifolds, tangent bundles, embeddings, immersions and submersions, Euler characteristic etc.
Essential Reading
- D. B. Gauld, Differential Topology: An Introduction, Dover Publication
- M. W. Hirsch, Differential Topology, Springer-Verlag
Supplementary Reading
- G. E. Bredon, Topology and Geometry, Springer-Verlag
- Amiya Mukherjee, Topics on Differential Topology, Hindustan Book Agency