Module I (20 hours)
Homotopy, Path homotopy, Homotopy equivalence, Contractible spaces, Fundamental group, Simply connected spaces, Retract, Deformation retract, Brouwer’s fixed point theorem, Covering map and covering space, Lifting theorems, Fundamental group of circle, Borsuk-Ulam theorem, Free group, Free products of groups, Seifert-Van Kampen theorem and its applications

Module II (20 hours)
Introduction to category theory, Affine sets and affine maps, Simplexes, Singular Homology group, reduced and relative homology groups, Chain complexes of free Abelian groups and homology groups, Short exact sequences of chain complexes, the five lemma, Mayer-Vietoris Sequence and its applications.

To understand the basic algebraic and geometric ideas involved in homotopy and homology theory.

Prove topological results by using algebraic methods.

Compute algebraic invariants associated to topological spaces and maps between them.

CO1. Students will learn about homotopy, path-homotopy and fundamental groups. They will also get to know, how to translate continuous maps between topological spaces to homomorphisms between corresponding fundamental groups and how the properties of maps are transformed from topological to algebraic settings.

CO2. Students will study characterizations and properties of contractible and simply-connected spaces. Also they will get familiar with homotopically equivalent spaces, retracts, deformation retracts and use of these concepts for calculation of fundamental groups.

CO3. They will learn about covering space theory and lifting theorems to determine standard fundamental groups, and application of Seifert-Van Kampen Theorem for computation of fundamental groups of complicated spaces by using standard fundamental groups. Also, they will study Brouwer's fixed point and Borsuk-Ulam theorems and applications of these results.

CO4. Students will study affine independent sets, affine and convex sets, affine maps and category theory. They will learn to apply the concepts of category theory to topology.

CO5. Students will learn about singular homology theory. They will be able to compute singular homology groups by using tools such as the Mayer-Vietoris sequence and exact sequences.

J. R. Munkres, Elements of Algebraic Topology, Addison Wesley Publishing Company

J. R. Munkres, Topology, Perason Publishing Inc.

J. J. Rotman, An Introduction to Algebraic Topology, Springer

A. Hatcher, Algebraic Topology, Cambridge University Press

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