Topology/Geometry II (MAT 531)

1. Differentiable manifolds and maps
   • Inverse and implicit function theorems
   • Submanifolds, immersions and submersions

2. The tangent bundle
   • Vector bundles, transition functions
   • Reconstruction of a vector bundle from transition functions
   • Equivalence classes of curves and derivations; tangent vectors
   • The tangent bundle of a manifold as a vector bundle, examples
   • Vector fields, differential equations and flows
   • Lie derivatives and bracket

3. Differential forms
   • Exterior differential, closed and exact forms
   • Distributions, foliations and Frobenius integrability theorem
   • Poincaré Lemma

4. Integration
   • Stokes' Theorem
   • Integration and volume on manifolds
   • De Rham cohomology
   • Chain and cochain complexes
   • Homotopy theorem
   • The degree of a map
   • The Mayer-Vietoris theorem

Typical references:

• Michael Spivak, A Comprehensive introduction to differential geometry,
  2^nd ed., Publish or Perish, Berkeley 1979;
• Glen Bredon,Topology and geometry,
  Springer-Verlag, 1993.