**Course Details:**

Homotopic maps, Homotopy equivalence and Homotopy type, Paths and Homotopy, Fundamental Group of the Circle, Induced Homomorphisms.

Free Products of Groups, Seifert–van Kampen theorem.

Covering spaces and Lifting Properties, Classification of Covering Spaces, Deck Transformations and Group Actions.

Simplices, Simplicial Complexes and Simplicial Maps, Abstract Simplicial Complexes, Simplicial Homology.

Singular Homology, Homotopy Invariance of Homology, Exact Sequences and Excision, Relative Homology Groups.

Mayer-Vietoris Sequences, Homology with Coefficients, Axioms for Homology.

Cohomology Groups, Universal Coefficient Theorem, Cohomology of Spaces.

Cup Product, Cohomology Ring, Kunneth Formula.

Poincare Duality, Orientations and Homology.

**Text Books:**

1. James R. Munkres, Elements of Algebraic Topology, CRC Press, 2018.

2. Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002.

3. William Fulton, Algebraic Topology: a First Course, Springer New York, 2013.

**Reference Books:**

1. Tammo tom Dieck, Algebraic Topology, European Mathematical Society, 2008.

2. William S. Massey, Algebraic Topology, an Introduction, Springer-Verlag.

3. Samuel Eilenberg, Norman Steenrod, Foundations of Algebraic Topology, Princeton University Press.

4. John Harper, Marvin Greenberg, Algebraic Topology: a First Course, CRC Press.

5. John M. Lee, Introduction to Topological Manifolds, Springer New York.