Notes on Spectral Sequences

1. Exact couples and spectral sequences 2 1.1. Spectral sequences 2 1.2. Exact couples 2 1.3. The spectral sequence associated to an exact couple 3 1.4. Example: the spectral sequence of a filtered space 3 References 4 2. Pages of the spectral sequence 4 2.1. Iterated subquotients 4 2.2. Formula for the rth page 5 2.3. Filtered spaces 6 3. Convergence 6 3.1. Simplest case: first quadrant spectral sequences 6 3.2. The E∞ page of a spectral sequence 6 3.3. E∞ for exact couples 7 3.4. The spectral sequence of a filtered space, E∞ page 7 3.5. Spectral sequences of filtered chain complexes and double complexes 8 3.6. Appendix: Detecting isomorphisms 9 References 10 4. Homology with local coefficients 10 4.1. Local coefficients 10 4.2. Path connected based spaces 11 4.3. Singular chains with local coefficients 11 4.4. Homology with local coefficients 11 4.5. Appendix: Detecting weak equivalences 12 5. The Serre spectral sequence 13 5.1. The coefficient system associated to a Serre fibration 13 5.2. The Serre spectral sequence: E page and convergence 14 5.3. The first differential and the E page 15 5.4. A canonical CW approximation 16 6. Examples 17 6.1. The loop space of a sphere 17 6.2. Homology of SU(3) 18 6.3. Trivial fibrations 18 6.4. The Hopf fibration 18 6.5. Euler characteristic 18 7. Cohomology spectral sequences 19 7.1. Cohomology spectral sequence of a filtered space 19 7.2. Cohomology with local coefficients 20 7.3. The cohomology Serre spectral sequence 20 8. Examples 20