L05 - Crash course in algebraic topology - Material

Lecture notes

Stephan Klaus: Crash course in algebraic topology, ICMU, Kyiv, 6-10 May 2024

Lecture notes will appear here soon.

I am very grateful to Dr. Kaveh Eftekharinasab (ICMU) for preparing the lecture notes!

Topics in the crash course

  1. Introduction: Fixed point theorems
  2. Basics on manifolds
  3. Homotopy
  4. CW complexes
  5. Mapping degree and applications
  6. Fibre bundles and fibrations
  7. Homology and cohomology
  8. McCord-Linearization of a space
  9. Classifying spaces

Keywords for the covered topics in the crash course:

1. Introduction: Fixed point theorems

  • Banach Fixed Point Theorem
  • Brouwer Fixed Point Theorem
  • Categories, functors and retracts
  • Homology functor
  • Proof of Brouwer Fixed Point Theorem

2. Basics on manifolds

  • Topological and smooth manifolds
  • Manifolds with boundary, bordism
  • Tangent bundle and vector bundles
  • Regular values and transversality
  • Projective spaces and associated bundles
  • Vector fields and flows
  • Lie groups and homogeneous spaces
  • Embeddings, knots and links
  • Whitney embedding theorem

3. Homotopy

  • What are good topological spaces?
  • Homotopy and homotopy groups
  • Cone, suspension, path space, loop space
  • Covering spaces
  • Chain complexes and exactness
  • Fibrations and long exact sequence
  • Cofibrations and Puppe long exact sequence

4. CW complexes

  • Glueing, cells and skeleta
  • Finite CW complexes and Euler characteristic
  • Examples
  • Cellular approximation
  • Theorem of Whitehead
  • Morse Theorie

5. Mapping degree and applications

  • Mapping degree and orientation
  • Homotopy and bordism invariance
  • Jordan-Brouwer Separation Theorem
  • Linking number
  • Theorem of Hopf
  • Theorem of Poincaré-Hopf

6. Fibre bundles

  • Fibre bundles and principal fibre bundles
  • Homogeneous fibre bundles
  • Hopf maps
  • Grassmannian, Stiefel and flag manifolds
  • Classical Lie groups as iterated spherical fibre bundles

7. Homology and cohomology

  • Unreduced and reduced homology theories
  • Singular homology
  • Cohomology and products
  • Mayer-Vietoris long exact sequence
  • Universal coefficient theorems and Künneth theorem
  • Cellular homology, applications and examples
  • Theorem of Whitehead
  • Poincaré duality
  • Thom isomorphism and Gysin long exact sequence

8. Linearization of a space

  • McCord construction RX
  • Properties
  • McCord homology
  • Eilenberg-MacLane spaces and McCord cohomology
  • Quick proofs of standard theorems
  • Theorem of James

9. Classifying spaces

  • Universal bundles and classifying spaces
  • Milnor and McCord BG
  • Examples
  • Compact Lie groups, maximal tori and cohomology of BG
  • Characteristic classes
  • Applications to embeddings
  • Applications to bordism