Over the last year spectral sequences seem to be being mentioned more and more, so I figured it was probably about time I worked out sort of what they're all about. I won't give the most generic definitions, I'll more aim to give an idea of how they work and then have a look at the Serre Spectral Sequence.

References for this entry are:

- Allen Hatcher - Spectral Sequences in Algebraic Topology,
- Wikipedia,
- Some notes I took during one of Craig's courses.

**Spectral Sequence Definition**

So what are these spectral sequences all about? Well according to Wikipedia: "In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalisation of exact sequences, and since their introduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory." Cool! I suppose a definition might be a good start:

**Definition:** A spectral sequence is a sequence of pairs, $\{(E^r_{*,*}, d_r)\}_{r \in \mathbb{Z}_{\geq r_0}}$ for some $r_0 \in \mathbb{Z}_{\geq 0}$, such that for $r \in \mathbb{Z}_{\geq r_0}$ we have

- if $p, q \in \mathbb{Z}$ then $E^r_{p,q}$ is an abelian group,
- if $p, q \in \mathbb{Z}$ then $d_r : E^r_{p, q} \rightarrow E^r_{p - r, q + r - 1}$,
- $d_r \circ d_r = 0$,
- $E^{r + 1}_{*,*} \cong H_* (E^r_{*,*}, d_r)$.

$E^r_{*,*}$ is called the $E^r$-page of the spectral sequence and the $d_r$s are called the differentials.

We tend to draw each page in a lattice, for example, the $E^1$-page looks like

Then the $E_2$ page looks like

We can see that on the $E^r$-page, the differentials go $r$ columns to the left and $r - 1$ rows up. To get the entries in the $E^{r + 1}$-page we take the homology of the chains in the $E^r$-page.

For $p, q \in \mathbb{Z}$, we say that $E^*_{p,q}$ converges if there exists $i_0 \in \mathbb{Z}_{>0}$ such that if $i \in \mathbb{Z}$ and $i > i_0$ then both $d_i^{p - i,q + i - 1} : E^i_{p, q} \rightarrow E^i_{p - i, q + i - 1}$ and $d_i^{p, q} : E^i_{p + i, q - i + 1} \rightarrow E^i_{p, q}$ are zero. It is easy to see that if this is the case, then for $i \in \mathbb{Z}$ and $i > i_0$ we have $E^i_{p,q} = E^{i + 1}_{p, q}$ (since the kernel of $d_i^{p - i, q + i - 1}$ is $E^i_{p, q}$ and the image of $d_i^{p, q}$ is zero). We call the resulting entry $E^\infty_{p, q}$ and say $E^r_{p, q}$ converges to $E^\infty_{p, q}$, written $E^r_{p, q} \Rightarrow E^\infty_{p, q}$. If $E^r_{p, q}$ converges at every $p, q \in \mathbb{Z}$ we say the spectral sequence converges to $E^\infty$.

**Serre Spectral Sequence**

The Serre spectal sequence relates the homologies of spaces in a fibration.

**Theorem:** Let $F \rightarrow X \rightarrow B$ be a fibration. Then there exists a spectral sequence with $E^2$-page $$E^2_{p,q} = H_p (B, H_q (F)),$$ which converges. Further, the homology of the total space ($X$) is then $$H_i (X) = \bigoplus_{p + q = i} E^\infty_{p, q}.$$

The fact that the spectral sequence converges is clear, as the only non-zero entries are in the first quadrant (positive $p$ and positive $q$). For a fixed $p, q \in \mathbb{Z}_{\geq 0}$, we can see that for $r > max(p, q+1)$ the differential to $E^r_{p, q}$ will be coming from a $0$ in the fourth quadrant and the differential from $E^r_{p,q}$ will be going to a zero in the second quadrant giving an isomorphism $E^r_{p, q} \cong E^{r + 1}_{p, q}$ etc.

Now we will look at an example calculation (Example 1.4, p9 Hatcher). We will look at a path space fibration $$\Omega B \rightarrow P B \rightarrow B,$$ where $\Omega B$ is the loop space of $B$ and $P B$ is the path space of $B$. We will look at the specific case when $B$ is a $K (\mathbb{Z}, 2)$, that is $\pi_2 (B) = \mathbb{Z}$ and $\pi_i (B) = 0$ for $i \neq 2$, $i > 0$. Now since $P B$ is the path space of $B$, it is therefore contractible. We also have that $\pi_i (\Omega B) = \pi_{i + 1} (B)$, and so $\Omega B$ is a $K (\mathbb{Z}, 1)$. Recalling that $S^1$ is a $K (\mathbb{Z}, 1)$ and $$H_* (S^1) = \left\{ \begin{array}{l l} \mathbb{Z}, & * = 0, 1,\\ 0, & * \geq 2, \end{array}\right.$$ we see that the only non-zero rows in the $E^2$-page ($E^2_{p, q} = H_p (B, H_q (\Omega B))$) will be $q=0$ and $q = 1$:

We can see that if $r > 2$, the differentials on the $E^r$-page move up at least $2$ rows and thus will all have to be zero as they either originate from or map to $0$. Furthermore, since $P B$ is contractible, it's homology is $$H_* (P B) = \left\{ \begin{array}{l l} \mathbb{Z}, & * = 0,\\ 0, & * \geq 1, \end{array}\right.$$ so the $E^\infty$-page of the spectral sequence must look like the following (since $H_i (P B) = \oplus_{p + q = i} E^\infty_{p, q}$):

Now we are able to calculate the homology of $K (\mathbb{Z}, 2)$. We have seen that the only possible non-zero differentials appear on the $E^2$-page and that the only entry in the $E^\infty$-page is in the $p=0$, $q=0$ position. Therefore, all the differentials shown above in the $E^2$ page must be isomorphisms in order to 'kill off' all the entries leaving the $E^3$-page the same as the $E^4$-page and the $E^5$-page etc, all the way up to the $E^\infty$-page. Having this in hand, we are easily able to calculate $H_* (K(\mathbb{Z}, 2))$.

$H_0 (K(\mathbb{Z},2))$ must be $\mathbb{Z}$, since the differentials to and from the $p = 0$, $q = 0$ entry on the $E^2$-page are zero giving $E^2_{0, 0} = E^3_{0,0} = E^\infty_{0,0}$. Knowing that all the other differentials are isomorphisms then gives us that $H_1 (K(\mathbb{Z},2)) = 0$, $H_2 (K(\mathbb{Z},2)) = H_0 (K(\mathbb{Z},2)) = \mathbb{Z}$, $H_3 (K(\mathbb{Z},2)) = H_1 (K(\mathbb{Z},2)) = 0$, etc. We end up with the following solution $$H_* (K(\mathbb{Z},2)) = \left\{ \begin{array}{l l} \mathbb{Z}, & * \text{ even},\\ 0, & * \text{ odd}. \end{array}\right.$$ At this point you may notice that this happens to be the same homology as $\mathbb{C} P^\infty$, and you'd be correct. Checking the Wikipedia page for Eilenberg–MacLane spaces you will see that $\mathbb{C} P^\infty$ is listed as a $K(\mathbb{Z},2)$.

There you go, that's a quick run down of what all this spectral sequence stuff is all about along with a nice example. Hopefully it's mostly correct, I make no guarantees!