Course Details
Subject {L-T-P / C} : MA5114 : Homotopy Theory { 3-0-0 / 3}
Subject Nature : Theory
Coordinator : Prof. Divya Singh
Syllabus
Brouwer's Fixed point theorem, Categories, Functors, Natural transformations, natural equivalence, Homotopy, convexity, contractibility, mapping cylinder and cones, paths and path connected spaces, Affine spaces, Affine maps, Homotopy as equivalence relation, Contractible Spaces, Homotopy of maps, Homotopy classes, Homotopically equivalent spaces with examples, Fundamental Groups, Induced maps and homomorphisms, Lifting property, Calculation of first homotopy groups, Function spaces, Group objects and cogroup objects, Loop space and suspension, Exact sequence of homotopy groups, Homotopy lifting property, Homotopy extension property, Fibrations and cofibrations, CW-complexes and their examples, attaching of maps, Homotopy groups of CW-complexes, The effect on the homotopy groups of a cellular extension, Spaces with prescribed homotopy groups, Weak homotopy equivalences and CW-approximation, Homotopy extension and classification theorems.
Course Objectives
- To give an idea of transforming topological problems into algebraic one.
- To introduce the applications of topology.
Course Outcomes
1. Students will learn about category theory and Algebraic topology. <br />2. Students will get familiar with the first Homotopy groups of some specific spaces and Homology groups.
Essential Reading
- J. J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag
- E. H. Spanier, Algebraic Topology, McGraw Hill
Supplementary Reading
- W. S. Massey, Algebraic Topology-An Introduction, Springer-Verlag
- G. E. Bredon, Topology and Geometry, Springer-Verlag