Differential Forms in Algebraic Topology, Spring 2024

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 246.
  • The lectures are from Week 1 through Week 17. The last class is on June 21, 2024.

References:

  • Raoul Bott & Loring W. Tu: Differential Forms in Algebraic Topology, 3rd corrected printing, Graduate Texts in Math., Vol. 82, Springer, 1995.
  • S.S. Chern & W.H. Chen: Lectures on Differential Geometry, Peking University Press, 1983.
  • Loring W. Tu: An Introduction to Manifolds, 2nd edition, Universitext, Springer, 2011.
  • Loring W. Tu:  Differential Geometry. Connections, Curvature, and Characteristic Classes, Graduate Texts in Math., Vol. 275, Springer, 2017.

Contents:

  • Differentiable manifolds (Review of Tu2011).
  • De Rham cohomology (Bott-Tu + Tu2011).
  • Connections and curvatures on vector bundles (Review of Chern-Chen + §10-11 of Tu2017).
  • Chern-Weil construction of characteristic classes (Bott-Tu + Tu2017).

Evaluation:

  • There will be about 5 homework assignments throughout the semester.
  • Evaluation will be based on homework only. No Final Exam.
  • Assignment for Homeworks 1–4 (pdf). Due date: June 22, 2024.
  • Assignment for Homework 5 (pdf). Due date: July 7, 2024.

Slides:

  • Smooth manifolds (pdf).
  • Real and complex projectives spaces (pdf).
  • Smooth maps on manifolds (pdf).
  • Tangent space (pdf).
  • Submanifolds (pdf).
  • Constant rank theorem. Immersions and submersions (pdf).
  • Tangent bundle (pdf).
  • Vector bundles (pdf).
  • Differential 1-forms (pdf).
  • Differential k-forms (pdf).
  • Exterior derivative (pdf).
  • Lie brackets, Lie derivative, interior multiplication (pdf).
  • Orientation of manifolds (pdf).
  • Manifolds with boundary (pdf).
  • Integration on manifolds (pdf).
  • Cochain complexes and cohomology (pdf).
  • De Rham cohomology (pdf).
  • Bump functions and partitions of unity (pdf; updated May 10, 2024).
  • Mayer-Vietoris sequence (pdf; updated May 22, 2024).
  • Homotopy invariance and Poincaré lemmas (pdf; updated May 22, 2024).
  • The Mayer-Vietoris argument and Poincaré duality (pdf; updated June 3, 2024).
  • Riemannian manifolds and affine connections (pdf).
  • Connections and curvature on vector bundles (pdf).
  • Chern-Weil construction of characteristic classes. Pontryagin classes (pdf).