Differential Forms in Algebraic Topology, Spring 2025

Lecturer:

  • Prof. Raphael Ponge.
  • E-mail: ponge[dot]math[at]icloud[dot]com.

Time and Location:

  • Fridays from 8:00-10:45, 3rd Teaching Building, Room 316.
  • The lectures are from Week 1 through Week 17. The last class is on June 20, 2024.

References:

  • Raoul Bott & Loring W. Tu: Differential Forms in Algebraic Topology, 3rd corrected printing, Graduate Texts in Math., Vol. 82, Springer, 1995 (e-book; free access from SCU campus).
  • Loring W. Tu: An Introduction to Manifolds, 2nd edition, Universitext, Springer, 2011 (e-book; free access from SCU campus).
  • Loring W. Tu:  Differential Geometry. Connections, Curvature, and Characteristic Classes, Graduate Texts in Math., Vol. 275, Springer, 2017 (e-book; free access from SCU campus).

Evaluation:

  • There will be about 5 homework assignments throughout the semester.
  • Evaluation will be based on homework only. No Final Exam.

Contents and Slides

  1. Differentiable Manifolds (Tu 2011):
    • Smooth manifolds (slides).
    • Real and complex projectives spaces (slides).
    • Smooth maps on manifolds (slides).
    • Tangent space (slides).
    • Submanifolds (slides).
    • Constant rank theorem. Immersions and submersions (slides).
    • Tangent bundle (slides).
    • Vector bundles (slides).
  2. Differential Forms (Bott-Tu + Tu2011):
    • Differential 1-forms.
    • Differential k-forms.
    • Exterior derivative.
    • Lie brackets, Lie derivative, interior multiplication.
    • Orientation of manifolds.
    • Manifolds with boundary.
    • Bump functions and partitions of unity.
    • Integration on manifolds.
  3. De Rham Cohomology (Bott-Tu + Tu2011)
    • Cochain complexes and cohomology.
    • De Rham cohomology.
    • Mayer-Vietoris sequence.
    • Homotopy invariance and Poincaré lemmas.
    • The Mayer-Vietoris argument and Poincaré duality.
  4. Connections and Curvature (Tu 2017)
    • Riemannian manifolds and affine connections.
    • Connections and curvature on vector bundles .
  5. Chern-Weil construction of characteristic classes (Bott-Tu + Tu 2017)
    • Chern-Weil construction of characteristic classes.
    • Pontryagin classes.
    • Chern classes.