2021-04-26_Research_Notes_1-21-2018

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 $$\{(E_r, d_r)\}_{r\in{\mathbb{Z}}}$$ 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 $${\mathbb{Z}}$$-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 $$E_2^{i,j} = H^i(B, H^j(F)) \Rightarrow H^{i+j}(E, {\mathbb{Q}})$$. (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

$$\xrightarrow{d_1^{-2,2}} E_1^{-1,2} \xrightarrow{d_1^{-1,2}}$$ $$E_1^{0,2} \xrightarrow{d_1^{0,2}}$$ $$E_1^{1,2}\xrightarrow{d_1^{1,2}}$$ $$E_1^{2,2}\xrightarrow{d_1^{2,2}}$$ $$E_1^{3,2}\xrightarrow{d_1^{3,2}}$$ $$E_1^{4,2} \xrightarrow{d_1^{4,2}}$$ $$E_1^{5,2} \xrightarrow{d_1^{5,2}}$$

$$\xrightarrow{d_1^{-2,1}} E_1^{-1,1}\xrightarrow{d_1^{-1,1}}$$ $$E_1^{0,1} \xrightarrow{d_1^{0,1}}$$ $$E_1^{1,1}\xrightarrow{d_1^{1,1}}$$ $$E_1^{2,1}\xrightarrow{d_1^{2,1}}$$ $$E_1^{3,1}\xrightarrow{d_1^{3,1}}$$ $$E_1^{4,1}\xrightarrow{d_1^{4,1}}$$ $$E_1^{5,1} \xrightarrow{d_1^{5,1}}$$

$$\xrightarrow{d_1^{-2,0}}E_1^{-1,0} \xrightarrow{d_1^{-1,0}}$$$$E_1^{0,0} \xrightarrow{d_1^{0,0}}$$$$E_1^{1,0} \xrightarrow{d_1^{1,0}}$$$$E_1^{2,0} \xrightarrow{d_1^{2,0}}$$$$E_1^{3,0}\xrightarrow{d_1^{3,0}}$$$$E_1^{4,0}\xrightarrow{d_1^{4,0}}$$$$E_1^{5,0} \xrightarrow{d_1^{5,0}}$$

$$\frac{\ker d_0^{-1,2}}{\operatorname{im}d_0^{-2,2}}$$ $$\frac{\ker d_0^{0,2}}{\operatorname{im}d_0^{-1,2}}$$ $$\frac{\ker d_0^{1,2}}{\operatorname{im}d_0^{0,2}}$$ $$\frac{\ker d_0^{2,2}}{\operatorname{im}d_0^{1,2}}$$ $$\frac{\ker d_0^{3,2}}{\operatorname{im}d_0^{2,2}}$$ $$\frac{\ker d_0^{4,2}}{\operatorname{im}d_0^{3,2}}$$ $$\frac{\ker d_0^{5,2}}{\operatorname{im}d_0^{4,2}}$$

$$\frac{\ker d_0^{-1,1}}{\operatorname{im}d_0^{-2,1}}$$ $$\frac{\ker d_0^{0,1}}{\operatorname{im}d_0^{-1,1}}$$ $$\frac{\ker d_0^{1,1}}{\operatorname{im}d_0^{0,1}}$$ $$\frac{\ker d_0^{2,1}}{\operatorname{im}d_0^{1,1}}$$ $$\frac{\ker d_0^{3,1}}{\operatorname{im}d_0^{2,1}}$$ $$\frac{\ker d_0^{4,1}}{\operatorname{im}d_0^{3,1}}$$ $$\frac{\ker d_0^{5,1}}{\operatorname{im}d_0^{4,1}}$$

$$\frac{\ker d_0^{-1,0}}{\operatorname{im}d_0^{-2,0}}$$$$\frac{\ker d_0^{0,0}}{\operatorname{im}d_0^{-1,0}}$$$$\frac{\ker d_0^{1,0}}{\operatorname{im}d_0^{0,0}}$$$$\frac{\ker d_0^{2,0}}{\operatorname{im}d_0^{1,0}}$$$$\frac{\ker d_0^{3,0}}{\operatorname{im}d_0^{2,0}}$$$$\frac{\ker d_0^{4,0}}{\operatorname{im}d_0^{3,0}}$$$$\frac{\ker d_0^{5,0}}{\operatorname{im}d_0^{4,0}}$$

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.

#todo #spectral-sequences