MA7890 Advanced Algebraic Topology
Description: Main goal of topology is to study the properties of 'objects'. Now-a-days many techniques from topology are applicable to various subjects for example in biology, physics and differential equations. This proposed course will help the students to grasp the some important connections among topology, algebra and geometry.
Homotopy Theory: (16/17 lectures) Operations on Spaces, Homotopy and criteria for homotopy equivalence, Homotopy lifting property. Paths and path homotopy, Fundamental groups and examples. Homotopy invariants. Free groups and their products, Van Kampen theorem and several applications. Covering spaces and Deck transformations, Classification of covering spaces.
Homology Theory: (16/17 lectures) Cell complexes and Delta-complexes, Simplicial and Singular homology, Relative homology groups, Exact sequences and Excision, Equivalence of Simplicial and Singular homology. Homotopy invariance and degrees, Cellular homology and applications, Mayer-Vietories sequences. More applications.
Cohomology Theory: (17/18 lectures) Cohomology groups of spaces, Cup product andÂ cohomologyÂ ring, SomeÂ Spaces with polynomial cohomology rings, A Kunneth formula. Orientations and homology of manifolds, Cap product and Poincare Duality. Some applications.
1. Allen Hatcher: Algebraic topology. Cambridge University Press, Cambridge, 2002.
1. James Munkres: Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984.
2. William Massey: A basic course in algebraic topology. Graduate Texts in Mathematics, 127. Springer-Verlag, New York, 1991.
3. Glen Bredon: Topology and geometry. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1993.
1. Basic general topology, group and ring theory