Notes on Spectral Sequences

1. Local coefficient systems 2 1.1. The fundamental groupoid 2 1.2. Coefficient systems 3 1.3. Path connected based spaces 4 1.4. Singular chains with local coefficients 4 1.5. Homology with local coefficients 5 1.6. Universal covers 7 2. Exact couples and spectral sequences 9 2.1. Spectral sequences 9 2.2. Exact couples 9 2.3. The spectral sequence associated to an exact couple 10 2.4. Example: the spectral sequence of a filtered space 10 References 11 3. Pages of the spectral sequence 11 3.1. Iterated subquotients 11 3.2. Formula for the rth page 11 3.3. Filtered spaces 12 4. Convergence 13 4.1. Simplest case: first quadrant spectral sequences 13 4.2. The E∞ page of a spectral sequence 13 4.3. E∞ for exact couples 13 4.4. The spectral sequence of a filtered space, E∞ page 14 4.5. Spectral sequences of filtered chain complexes and double complexes 15 4.6. Appendix: Detecting isomorphisms 16 References 17 5. The Serre spectral sequence 17 5.1. The coefficient system associated to a Serre fibration 17 5.2. The Serre spectral sequence: E page and convergence 18 5.3. The first differential and the E page 18 5.4. A canonical CW approximation 19 6. Examples 21 6.1. The loop space of a sphere 21 6.2. Homology of SU(3) 21 6.3. Trivial fibrations 22 6.4. The Hopf fibration 22 6.5. Euler characteristic 22 7. Cohomology spectral sequences 22 7.1. Cohomology spectral sequence of a filtered space 22 7.2. Cohomology with local coefficients 23