Syllabus for the Comprehensive Exam in Topology

  1. Smooth Maps: The inverse and implicit function theorems; immersion; submersion; embedding; regular values; submanifolds arising as point preimages; Sard's theorem; Whitney's embedding theorem
  2. Smooth Manifolds: Topological manifolds; differentiable manifolds; submanifolds and examples
  3. Tangent Spaces: Differentials; tangent bundles; vector fields; flows/integrating vector fields; Lie bracket and Lie derivative
  4. Transversality: Transversality; intersection theory; degrees of maps; winding number; the Poincaré-Hopf theorem
  5. Differential forms: tensors, differential forms, exterior derivative, integration on manifolds, basic de Rham cohomology
  6. The Fundamental Group: Homotopy; the van Kampen theorem; examples
  7. Covering Spaces: Path and homotopy lifting; general lifting theorem for maps; universal covers; regular covers; deck transformations; correspondence between subgroups and covers; computing the fundamental group via covering spaces


Suggested textbooks: A Comprehensive Introduction to Differential Geometry, Vol. 1 by Spivak; Differential Topology by Guillemin and Pollack; Algebraic Topology by Hatcher
Suggested courses: 6452 and 6441
Other relevant courses: 4431