L06 - Lectures on RX - Material

Topics and keywords (preliminary)

1. What are good topological spaces?

  • Simplicial complexes
  • Manifolds
  • CW complexes
  • Compactly generated weak Hausdorff spaces
  • Simplicial sets
  • Cellular approximation

2. Homology theories

  • Axioms of a unreduced homology theory
  • Axioms of a reduced homology theory
  • Cellular homology
  • Corollary: uniqueness for CW complexes

3. Singular homology and cohomology

  • Definition
  • On the proof of the axioms
  • Mayer Vietoris LES
  • Universal coefficient theorem
  • Künneth theorem

4. McCord linearization RX of a space

  • Definition of RX
  • Basic properties
  • Connection to symmetric products

5. Classifying space BG

  • Definition by Milnor 
  • Definition by McCord
  • Remark: group completion theorem

6. McCord homology

  • Definition of H*(X,Y;R)
  • Proof of the axioms
  • Corollary: Dold-Thom theorem

7. Quick proofs of many standard results

  • Proof of Mayer-Vietoris LES and SPS
  • Proof of Bockstein LES
  • Proof of universal coefficient theorems
  • Proof of Künneth theorem

8. Eilenberg-MacLane spaces, McCord cohomology and products

  • K(R,n) = RSn is an abelian topological group
  • Definition of H*(X,Y;R)
  • Products and their properties

9. Bott-Samelson Theorem

  • James construction JX and JX ~ ΩSX
  • Bott-Samelson theorem
  • James-Milnor splitting
  • Freudenthal suspension theorem

10. Twisted coefficients and Serre spectral sequence

  • Twisted coefficients
  • Twisted tensor products
  • Serre spectral sequence

11. Characteristic classes

  • Theorem of Borel
  • Topological polynomial ring on a space
  • Chern classes

12. Problems and work in progress

  • Proof of Whitehead theorem?
  • Proof of Bott periodicity?
  • Explicit construction of cohomology operations?