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Programmes → I.M.Sc. in Mathematical Sciences; M.Sc. Mathematics/Applied Mathematics

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Topology-II

Course Code: MA 482; MM 577; AM 577  |  Credits: 4  |  Prerequisite: Topology-I, Algebra-I

Syllabus Details

  • Unit I: (The Fundamental groups and homotopy theory) Homotopy mappings, contractible spaces, homotopically equivalent spaces, fundamental group of topological spaces, homotopy invariance of the fundamental group, the fundamental group of a circle is isomorphic to the group of integers. The Brower’s fixed point theorem, the fundamental theorem of algebra, introduction to higher homotopy groups.
  • Unit II: (The category theory) Category theory and the functors between the categories, the fundamental group functor.
  • Unit III: (The Van Kampen theorem and CW-complexes) Finding a topological space for a given group as a fundamental group, Van Kampen theorem to calculate the fundamental group of a topological space when fundamental groups of subspaces are given.
  • Unit IV: (Covering space theory) Definition of a covering Space and examples of covering spaces. Properties of covering spaces, the relation between the fundamental group of covering space and its base, the lifting criteria, universal covering space construction.
  • Unit V: (Simplicial theory): Various notions like simplicial complexes, barycentric subdivision, simplicial maps, approximation theorem, the fundamental group of a simplicial complex.

Course Learning Objective

  1. Understand the homotopy theory.
  2. Classify a few topological spaces up to the homotopy equivalence.
  3. Calculate the fundamental groups of topological spaces.
  4. State and prove the Van Kampen theorem and understand CW-complex.
  5. Construct a topological space whose fundamental group is the given group.
  6. Understand the covering spaces and examples.
  7. Apply covering theory to the lifting problems, a few questions in complex analysis.
  8. Define various concepts from homological algebra.
  9. Calculate different homologies for some topological space.
  10. A student should get familiar with the category theory.
  1. Bredon, G. E., Topology and Geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1997.
  2. Hatcher, A., Algebraic Topology, Cambridge University Press, 2002.
  3. Hocking, J. G. and Young, G. S., Topology, second edition, Dover Publications, New York, 1988.
  4. Greenberg, M. J. and Harper, J. R., Algebraic Topology: A First Course, Mathematics Lecture Note Series 58, Benjamin/Cummings Publishing Co., 1981.
  5. Massey, W. S., A Basic Course in Algebraic Topology, Graduate Texts in Mathematics 127, Springer-Verlag, 1991.
  6. Maunder, C. R. F., Algebraic Topology, Cambridge University Press, 1980.
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