Algebraic Topology | Department of Mathematics

Algebraic Topology

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 640, MAT 622. Undergraduates can substitute MAT 640 with MAT 240.

Overview: Algebraic topology is a tool which captures key information about geometrical properties of a topological space. We shall focus on two basic algebraic invariants of topological spaces: the fundamental group and the simplicial homology groups. If time permits, we shall also get a glimpse of singular homology.

Detailed Syllabus:

  1. The Fundamental Group
    • Homotopy, Fundamental Group, Introduction to Covering Spaces, The Fundamental Group of the circle S 1 , Retractions and fixed points, Application to the Fundamental Theorem of Algebra, The Borsuk-Ulam Theorem, Homotopy Equivalence and Deformation Retractions, Fundamental group of a product of spaces, torus, n-sphere, and the real projective n-space.
    • van Kampen’s Theorem: Free Products of Groups, The van Kampen Theorem, Fundamental Group of a Wedge of Circles, Definition and construction of Cell Complexes, Application of Van Kampen Theorem to Cell Complexes, Statement of the Classification Theorem for Surfaces.
  2. Covering Spaces: Universal Cover and its existence, Unique Lifting Property, Homomorphisms and automorphisms of Covering Spaces, Action of the fundamental group on the fibers, Deck Transformations, Group Actions, Covering Space Actions, Normal or Regular Covering Spaces.
  3. Simplicial Homology (If time permits): Finite Simplicial complexes, Polyhedra and Triangulations, Simplicial approximation, Barycentric subdivision.  Orientation of simplicial complexes, Simplical chain complex and homology. Invariance of homology groups. Computations and applications.

References:

  1. A.Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002.
  2. S. Deo, Algebraic Topology, Hindustan Book Agency, 2006.
  3. W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, Berlin, 1991.
  4. W. Fulton, Algebraic topology: A First Course, Springer-Verlag, 1995.
  5. J.R. Munkres, Elements of Algebraic Topology, Addison Wesley, 1984.
  6. J.J. Rotman, An Introduction to Algebraic Topology, Springer (India), 2004.
  7. J.W. Vick, Homology Theory, Springer-Verlag, 1994.
  8. E. H. Spanier, Algebraic Topology, Springer, 1994.
Course Code: 
MAT722
Course Credits: 
4.00
Department: 
Course Level: