Overview Edit

I intend on giving a series (if 'series' can be applied, hopefully not too many) of seminars that build up to spectral sequences, since I suspect them to be too inaccessible to hit in one talk with any amount real of satisfaction. In fact, so far as I can judge, the usual crowd has had little to no experience with algebraic topology whatsoever. May as well start from the top.

Check back here for transcripts, the first should occur sometime mid-August.

Topics Edit

Categories and Topology Edit


  • Brief review of algebra: groups, rings, exact sequences
  • Categories: functors, co(ntra)variance, and (co)limits
  • Realization of Top: product and coproduct, coequalizer, pushout
  • Top*, the smash product, and the exponential law

Homotopy and Fibrations Edit


  • Applications of the smash: cones, suspensions, loop spaces
  • Homotopy and the pathspace
  • Group objects and H-(co)groups
  • Induced algebraic structures on hom-sets, the homotopy groups of a space
  • Relative homotopy and its long exact sequence
  • Fibrations, fibered spaces, covering spaces

Cellular Complexes and Ordinary Homology Edit


  • Simplicial and cellular complexes
  • Homotopy properties of cell complexes:
    • Cellular / simplicial approximation
    • Groups stable under cell attachment
    • Homotopy groups of coproducts
    • Cofibrations
    • Whitehead's theorem
    • Building spaces from homotopy groups
    • Eilenberg-Maclane spaces and connective fibrations, Postnikov towers
  • Introduction to cellular homology and its dual cohomology

Spectra and Spectral Sequences Edit


  • Homological tidbits
    • The Mayer-Vietoris long exact sequence
    • Kunneth's formula, the cup product in ordinary cohomology
  • Spectra
    • The category of spectra
    • Suspension and its inverse
    • Cohomology associated to a spectrum
    • Spectrum associated to a cohomology
    • Products and ring spectra
  • Spectral sequences
    • Exact couples, convergence
    • Filtrations, Atiyah-Hirzebruch spectral sequence
    • A sketch of the Serre spectral sequence
    • Computations using the Serre spectral sequence, common fibrations