Topics

Scheduling

Fridays 12-2 PM EST

  • Week 1 (May 21): Preliminary Review (Point-Set)
    • Topologies, continuity, homeomorphisms, subspaces and products, closures, open/closed/clopen, bases, retracts
    • Compactness, metric spaces, completeness, boundedness
    • Connectedness (path-connected & “locally” versions), totally disconnected
    • mSeparation axioms, Hausdorff spaces, normal, regular
    • The tube lemma
    • Common counterexamples (sine curve)
  • Week 2 (May 38): The Fundamental Group
    • Van Kampen
    • Homotopic maps vs conjugacy, change of base point
    • Nullhomotopic maps
  • Week 3 (June 4): Covering Spaces
    • Definitions, HLP, deck transformations, regular covers, universal cover
    • Classifications of covers, applications to subgroups of free groups
    • Lifting criterion, Galois correspondence
  • Week 4 (June 11): CW Complexes, Pushouts
    • Cellular chain complex
    • Euler characteristic
    • Hurewicz, LES in homotopy
    • Simplicial complexes
  • Week 5 (June 18): Homology
    • Mayer-Vietoris, Kunneth, UCT
    • Cellular homology (direct definition)
    • Relative homology
    • Facts for manifolds (dim 3 and 4), Poincare duality
  • Week 6 (June 25): Surfaces
    • Definition of manifolds (sphere, torus, projective spaces)
    • Connect sum, fundamental polygons and gluing
    • Classification of surfaces, boundary components
  • Week 7 (July 2): Fixed Points & Degree Theory
    • Degree theory
    • Lefschetz, Brouwer, Hairy Ball Theorem, Borsuk-Ulam
  • Week 8 (July 9): Buffer
  • Week 9 (July 16): Buffer
  • Week 10 (July 23): Buffer
  • Week 11 (July 30): Buffer
  • Week 12 (August 6): Buffer
  • Week 13 (August 13): Timed practice exam
  • Quals: Monday and Tuesday, August 16-17

Topics

  • Fundamental groups
  • Covering spaces
  • Higher homotopy groups.
  • Fibrations and the long exact sequence of a fibration
  • Singular homology and cohomology
  • Relative homology
  • CW complexes and the homology of CW complexes.
  • Mayer-Vietoris
  • Universal coefficient theorem
  • Kunneth formula
  • Poincare duality
  • Lefschetz fixed point formula
  • Hopf index theorem
  • Cech cohomology and de Rham cohomology.
  • Equivalence between singular, Cech and de Rham cohomology