Spectral Sequences and Topological Covers

Abstract

Given a space $X$ and a topological cover $\mathscr U = \{U_\lambda\}_{\lambda \in \Lambda}$ we can construct a spectral sequence that can be used to calculate the homology of $X$, called the Mayer-Vietoris Spectral Sequence. The cover also gives rise to a homotopy colimit spectral sequence (also known as the Bousfield-Kan Spectral Sequence) that can be shown to coincide with the Mayer-Vietoris spectral sequence for special cases. In particular, we can recover the Mayer-Vietoris long exact sequence in singular homology from both. A leisurely introduction to spectral sequences will be given, showing how to use them for basic diagram chases, as well as how to balance Ext and Tor.