Introduction to Algebraic Topology

ABOUT THE COURSE:This is an introductory course in Algebraic Topology. In this course we will study some algebraic invariants associated to Topological Spaces. In particular, we will focus on two such invariants: Fundamental group and Homology groups. We will also learn about covering spaces which is a very important and useful topic in mathematics. We will focus both on conceptual understanding and computational techniques. We will also see various applications of these invariants in different areas of mathematics.INTENDED AUDIENCE: MSc/PhD Mathematics StudentsPREREQUISITES: Linear AlgebraAbstract AlgebraReal AnalysisPoint-Set-Topology

Syllabus

Week 1: Recalling the basic concepts from point set topology: Topological spaces, continuous functions, topological invariants e.g., compactness, connectedness etc.
Homotopy and Homotopy type.
Week 2:Simplicial complexes and Homotopy
Week 3:The Fundamental group. Induced Homomorphisms.
Week 4:The Fundamental Group of Circle.
Week 5:Theory of Covering Spaces
Week 6:Free group and Seifert Van Kampen Theorem
Week 7:Simplicial Complexes and Delta Complexes
Week 8:Simplicial Homology: Definitions and Basic Properties.
Week 9:Some basics of Homological Algebra.
Week 10:Exact Sequences and Excision
Week 11:Mayer Vietoris Sequence
Week 12:Applications of Homology.