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Indian Institute of Science Education and Research Berhampur

Mathematics

MTH 516: Topology II (4)

Pre-requisites (Desirable): MTH 507 or MTH 605 and MTH 302 or MTH 601

Learning Objectives:

This is an advanced course in Topology.

Course Contents:

Simplicial Homology: Simplicial Complexes, Barycentric Subdivision, and Simplicial Homology with examples  

Singular and Cellular Homology: Definition with examples, Homotopy Invariance, Exact Sequence of Relative Homology, Excision, Mayer-Vietoris Sequence, Degree of Maps, and Cellular Homology, Jordan-Brouwer Separation Theorem, Invariance of domain and dimension, Borsuk-Ulam Theorem,  Lefschetz-Hopf Fixed Point Theorem, Axioms for homology, Fundamental group and homology, and Simplicial Approximation Theorem

Cohomology: Universal Coefficient Theorem, Künneth Formula, Cup Product and the Cohomology Ring, Cap Product, Orientations on Manifolds, and Poincaré Duality

Higher Homotopy Groups: Definition with examples, Aspherical Spaces, Relative Homotopy Groups, Long Exact Sequence of  a triple, n-connected spaces, and Whitehead's Theorem

Suggested Books:


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