Stuff popping up everywhere:
- Pushforwards
- Derived functors (a little)
- Projective/Injective resolutions
Motto for homology: kernel of what’s going out mod image of what’s coming in
Easy definition: A spectral sequence is the data \(\{(E_r, d_r)\}_{r\in{\mathbf{Z}}}\) where each \(E_r\) is an abelian group, \(d_r: E_r \to E_r\) is a homomorphism satisfying \(d_r^2=0\), and \(E_{r+1} \cong \frac{\ker d_r}{\operatorname{im}d_r}\).
Another definition: a homological spectral sequence is a sequence of \({\mathbf{Z}}\)-bigraded modules \(\{E^r_{p,q}\}_{r > 0}\) with differentials \(d_r: E^r_{p,q} \to E^r_{p-r, q+(r-1)}\) such that \(E^{r+1} = H_*(E^r)\).
A cohomological spectral sequence is the same, except \(d_r: E_r^{p,q} \to E_r^{p+r, q-(r-1)}\)
The ‘lines’ with slope \(-\frac{r-1}{r}\) form chain complexes.
Define cycles to be \(Z_i \coloneqq\ker d_i\), boundaries to be \(B_i: \operatorname{im}d_i\).
Concrete examples for pages:
\(r=1\): Differential is \(d_1:E^1_{p,q} \to E^1_{p-1, q}\)
\(r=2\): Differential is \(d_2:E^2_{p,q} \to E^2_{p-2, q+1}\)
Equivalently, \(d_2: H_*(E^1_{p,q}) \to H_*(E^1_{p-1, q})\)?
\(r=3\): Differential is \(d_3: E^3_{p,q} \to E^3_{p-3,q+2}\)
Should be able to compute the cohomology rings of fiber bundles \(E \xrightarrow{f} B\) pretty easily, using the map induced by the cup product \(E_r^{i,j} \times E_r^{k,l} \to E_r^{i+k, j+l}\) and the fact that \(E_2^{i,j} = H^i(B, H^j(F)) \Rightarrow H^{i+j}(E, {\mathbf{Q}})\). (For example, try \(SO_{n-1} \to SO_n \to S^{n-1}\))
How to put a filtration in the \(E^1\) page: ?
Any complex with a two step filtration \(F_1 \subset F_0 = K\) is exactly the long exact arising from \(0 \hookrightarrow F^1 \hookrightarrow F_0 \twoheadrightarrow\frac{F_1}{F_0} \twoheadrightarrow 0\).
Next simplest example: a three step filtration \(F_2 \subset F_1 \subset F_0 = K\). Write down all of the short exact sequences, and relate \(H^*(K)\) to \(H^*(\frac{F^i}{F^{i+1}})\).
Index Reference
The \(E_0\) page
\(\xrightarrow{d_0^{-2,2}} E_0^{-1,2} \xrightarrow{d_0^{-1,2}}\) \(E_0^{0,2} \xrightarrow{d_0^{0,2}}\) \(E_0^{1,2}\xrightarrow{d_0^{1,2}}\) \(E_0^{2,2}\xrightarrow{d_0^{2,2}}\) \(E_0^{3,2}\xrightarrow{d_0^{3,2}}\) \(E_0^{4,2} \xrightarrow{d_0^{4,2}}\) \(E_0^{5,2} \xrightarrow{d_0^{5,2}}\)
\(\xrightarrow{d_0^{-2,1}} E_0^{-1,1}\xrightarrow{d_0^{-1,1}}\) \(E_0^{0,1} \xrightarrow{d_0^{0,1}}\) \(E_0^{1,1}\xrightarrow{d_0^{1,1}}\) \(E_0^{2,1}\xrightarrow{d_0^{2,1}}\) \(E_0^{3,1}\xrightarrow{d_0^{3,1}}\) \(E_0^{4,1}\xrightarrow{d_0^{4,1}}\) \(E_0^{5,1} \xrightarrow{d_0^{5,1}}\)
\(\xrightarrow{d_0^{-2,0}}E_0^{-1,0} \xrightarrow{d_0^{-1,0}}\) \(E_0^{0,0} \xrightarrow{d_0^{0,0}}\) \(E_0^{1,0} \xrightarrow{d_0^{1,0}}\) \(E_0^{2,0} \xrightarrow{d_0^{2,0}}\) \(E_0^{3,0}\xrightarrow{d_0^{3,0}}\) \(E_0^{4,0}\xrightarrow{d_0^{4,0}}\) \(E_0^{5,0} \xrightarrow{d_0^{5,0}}\)
\(\xrightarrow{d_1^{-2,2}} E_1^{-1,2} \xrightarrow{d_1^{-1,2}}\) \(E_1^{0,2} \xrightarrow{d_1^{0,2}}\) \(E_1^{1,2}\xrightarrow{d_1^{1,2}}\) \(E_1^{2,2}\xrightarrow{d_1^{2,2}}\) \(E_1^{3,2}\xrightarrow{d_1^{3,2}}\) \(E_1^{4,2} \xrightarrow{d_1^{4,2}}\) \(E_1^{5,2} \xrightarrow{d_1^{5,2}}\)
The \(E_1\) page
\(\xrightarrow{d_1^{-2,2}} E_1^{-1,2} \xrightarrow{d_1^{-1,2}}\) \(E_1^{0,2} \xrightarrow{d_1^{0,2}}\) \(E_1^{1,2}\xrightarrow{d_1^{1,2}}\) \(E_1^{2,2}\xrightarrow{d_1^{2,2}}\) \(E_1^{3,2}\xrightarrow{d_1^{3,2}}\) \(E_1^{4,2} \xrightarrow{d_1^{4,2}}\) \(E_1^{5,2} \xrightarrow{d_1^{5,2}}\)
\(\xrightarrow{d_1^{-2,1}} E_1^{-1,1}\xrightarrow{d_1^{-1,1}}\) \(E_1^{0,1} \xrightarrow{d_1^{0,1}}\) \(E_1^{1,1}\xrightarrow{d_1^{1,1}}\) \(E_1^{2,1}\xrightarrow{d_1^{2,1}}\) \(E_1^{3,1}\xrightarrow{d_1^{3,1}}\) \(E_1^{4,1}\xrightarrow{d_1^{4,1}}\) \(E_1^{5,1} \xrightarrow{d_1^{5,1}}\)
\(\xrightarrow{d_1^{-2,0}}E_1^{-1,0} \xrightarrow{d_1^{-1,0}}\) \(E_1^{0,0} \xrightarrow{d_1^{0,0}}\) \(E_1^{1,0} \xrightarrow{d_1^{1,0}}\) \(E_1^{2,0} \xrightarrow{d_1^{2,0}}\) \(E_1^{3,0}\xrightarrow{d_1^{3,0}}\) \(E_1^{4,0}\xrightarrow{d_1^{4,0}}\) \(E_1^{5,0} \xrightarrow{d_1^{5,0}}\)
\(\frac{\ker d_0^{-1,2}}{\operatorname{im}d_0^{-2,2}}\) \(\frac{\ker d_0^{0,2}}{\operatorname{im}d_0^{-1,2}}\) \(\frac{\ker d_0^{1,2}}{\operatorname{im}d_0^{0,2}}\) \(\frac{\ker d_0^{2,2}}{\operatorname{im}d_0^{1,2}}\) \(\frac{\ker d_0^{3,2}}{\operatorname{im}d_0^{2,2}}\) \(\frac{\ker d_0^{4,2}}{\operatorname{im}d_0^{3,2}}\) \(\frac{\ker d_0^{5,2}}{\operatorname{im}d_0^{4,2}}\)
\(\frac{\ker d_0^{-1,1}}{\operatorname{im}d_0^{-2,1}}\) \(\frac{\ker d_0^{0,1}}{\operatorname{im}d_0^{-1,1}}\) \(\frac{\ker d_0^{1,1}}{\operatorname{im}d_0^{0,1}}\) \(\frac{\ker d_0^{2,1}}{\operatorname{im}d_0^{1,1}}\) \(\frac{\ker d_0^{3,1}}{\operatorname{im}d_0^{2,1}}\) \(\frac{\ker d_0^{4,1}}{\operatorname{im}d_0^{3,1}}\) \(\frac{\ker d_0^{5,1}}{\operatorname{im}d_0^{4,1}}\)
\(\frac{\ker d_0^{-1,0}}{\operatorname{im}d_0^{-2,0}}\) \(\frac{\ker d_0^{0,0}}{\operatorname{im}d_0^{-1,0}}\) \(\frac{\ker d_0^{1,0}}{\operatorname{im}d_0^{0,0}}\) \(\frac{\ker d_0^{2,0}}{\operatorname{im}d_0^{1,0}}\) \(\frac{\ker d_0^{3,0}}{\operatorname{im}d_0^{2,0}}\) \(\frac{\ker d_0^{4,0}}{\operatorname{im}d_0^{3,0}}\) \(\frac{\ker d_0^{5,0}}{\operatorname{im}d_0^{4,0}}\)
Differentials
Recovering the homology
If a spectral sequence collapses, say \(E_\infty^{p,q} = E_N^{p,q}\), then \(H_n(X)\) is the unique \(E_N^{p,q}\) where \(p+q=n\). In general, the homology can be read off as the single nonzero element on the diagonal when this happens.