Homology Serre spectral sequence for the Hopf fibration
The Serre spectral sequence can be constructed by choosing a cell structure on the base space, and looking at the induced filtration on the total space. In the case of the Hopf fibration $p: S^3 \to S^2$, and making use of the easy cell structure on S^2 with only two cells, we obtain the following filtration:
$X_k = \emptyset$ for $k \leq 0$
$X_0 = X_1 = S^1$
$X_k = S^3$ for $k \geq 2$
The homology exact couple of a filtered space is built from the various long exact sequences associated to the pairs $(X_p , X_{p-1})$.
Its first page is the relative homology, and its first auxiliary is the (absolute) homology. The variables act as follows:
$f : H_n X_p \to H_n X_{p+1}$
$e : H_n X_p \to H_n(X_p , X_{p-1})$
$e : H_n(X_p , X_{p-1}) \to H_{n-1} X_{p-1}$,
where this last map is the usual connecting homomorphism. In our example, the most interesting long exact sequence concerns the pair $(S^3,S^1)$. Since the homology of $S^3$ and $S^1$ are pretty sparse, the homology of the pair is determined by isomorphisms
$e : H_3 S^3 \to H_3(S^3,S^1)$ since $H_3$ and $H_2$ vanish on $S^1$
$e : H_2(S^3,S^1) \to H_1 S^1$ since $H_2$ and $H_1$ vanish on $S^3$
$H_1(S^3,S^1) = 0$ since $H_1 S^3 = 0$ and $H_0 S^1 \to H_0 S^3$ is an injection
$H_0(S^3,S^1) = 0$ since $H_0 S^1 \to H_0 S^3$ is a surjection.
This analysis shows that the four groups $H_0 X_0$, $H_1 X_0$, $H_2(X_2, X_1)$, and $H_3 X_2$ are free abelian of rank one. Let $x \in H_0 X_0$, $y \in H_1 X_0$, $z \in H_2(X_2,X_1)$, and $w \in H_3 X_2$ be generating classes for these groups.
By the general properties of exact couples, we have that $e^2$ annihilates auxiliary generators and $f$ annihilates page generators. It turns out that the only remaining relation in the exact couple is $e*z - f*y$, which says that the connecting map $H_2(X_2,X_1) \to H_1(X_1)$ sends $z$ to the same place as the filtration map $H_1(X_0) \to H_1(X_1)$ sends $y$. (Depending on your choices for generators, this relation may read $e*z + f*y$.)
It is straightforward to give all this information to M2, but determining degrees does take a bit of thought the first time you do it. We continue the example by analyzing degrees ...
Since we have specific degree preferences (exactly double the usual ones for a Serre spectral sequence) we will do this by hand rather than relying on the default degrees provided by coupleRing.
The usual Serre spectral sequence has $E^1_{pq} = H_{p+q}(X_p , X_{p-1})$, so we place this module in degree $(2p,2q)$. The module $A^1_{pq} = H_{p+q} X_p$ sits halfway along the map $E^1_{pq} \to E^1_{(p-1)q}$, so it has degree $(2p-1,2q)$.
In usual indexing, the differential has degree $(-1,0)$, so our differential has degree $(-2,0)$. On the other hand, since the differential is given by $e^2$, this means that the degree of $e$ itself is $(-1,0)$. (And in general, the degree of e in our conventions should be the degree of the differential in the usual conventions.)
Everything so far has been about the first page, $E^1$. However, the easiest way to determine the degree of $f$ is to consider the second page. The usual conventions give that the differential on $E^2$ has degree $(-2,1)$, and so (by the same argument as above) this is the degree of $e_2$. We have generally $deg e_{k+1} = deg e_k - (deg f_k)/2$. Solving, we see that the degree of $f = f_1$ is $2 ((-1,0) - (-2,1)) = (2,-2)$.
We conclude the general rule that... the degree of e_k in our convention equals the degree of the page-k differential in the standard convention; the degree of f_k is then given by * (deg e_k - deg e_{k+1})$.
We are now able to set up the couple ring.
i1 : R = QQ; |
i2 : Q = R[e_1,f_1,Degrees=>{{-1,0},{2,-2}}];
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Now it is time to build the couple. We give our couple ring to the function declareCouple, together with names and degrees for our generating classes. The page generators are always given first, and the auxiliary generators second.
i3 : declareCouple(Q, {z => {4,0}}, {x => {1,0}, y => {1,2}, w => {5,2}})
o3 = cokernel {5, 2} | e_1^2 e_1f_1 0 0 0 0 0 0 |
{1, 0} | 0 0 e_1^2 e_1f_1 0 0 0 0 |
{1, 2} | 0 0 0 0 e_1^2 e_1f_1 0 0 |
{4, 0} | 0 0 0 0 0 0 e_1^3 f_1 |
4
o3 : Q-module, quotient of Q
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The couple we have built is "free" in the sense that the only relations imposed are the tautologous ones that hold in every exact couple (f acts by zero on the page, and e^2 acts by zero on the auxiliary). To obtain our couple, we must impose the relation $e*z-f*y$.
i4 : C = cospan(e_1*z-f_1*y)
o4 = cokernel {5, 2} | 0 e_1^2 e_1f_1 0 0 0 0 0 0 |
{1, 0} | 0 0 0 e_1^2 e_1f_1 0 0 0 0 |
{1, 2} | -f_1 0 0 0 0 e_1^2 e_1f_1 0 0 |
{4, 0} | e_1 0 0 0 0 0 0 e_1^3 f_1 |
4
o4 : Q-module, quotient of Q
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i5 : isHomogeneous C o5 = true |
i6 : expectExactCouple C |
Since expectExactCouple accepts C without an error, we truly have an exact couple, and are ready to compute its spectral sequence.
The next line displays the some entries in the first four pages of the spectral sequence determined by C.
i7 : plotPages((-2..4,-2..3,1..4), prune @@ evaluateInDegree,C)
page 1, with differential of degree {-1, 0}:
+----++----+----+---+---+---+---+---+
|q=3 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
| || | | 1| | 1| | |
|q=1 ||0 |0 |QQ |0 |QQ |0 |0 |
+----++----+----+---+---+---+---+---+
| || | | 1| | 1| | |
|q=0 ||0 |0 |QQ |0 |QQ |0 |0 |
+----++----+----+---+---+---+---+---+
|q=-1||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=-2||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
| ||p=-2|p=-1|p=0|p=1|p=2|p=3|p=4|
+----++----+----+---+---+---+---+---+
page 2, with differential of degree {-2, 1}:
+----++----+----+---+---+---+---+---+
|q=3 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
| || | | 1| | 1| | |
|q=1 ||0 |0 |QQ |0 |QQ |0 |0 |
+----++----+----+---+---+---+---+---+
| || | | 1| | 1| | |
|q=0 ||0 |0 |QQ |0 |QQ |0 |0 |
+----++----+----+---+---+---+---+---+
|q=-1||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=-2||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
| ||p=-2|p=-1|p=0|p=1|p=2|p=3|p=4|
+----++----+----+---+---+---+---+---+
page 3, with differential of degree {-3, 2}:
+----++----+----+---+---+---+---+---+
|q=3 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
| || | | | | 1| | |
|q=1 ||0 |0 |0 |0 |QQ |0 |0 |
+----++----+----+---+---+---+---+---+
| || | | 1| | | | |
|q=0 ||0 |0 |QQ |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=-1||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=-2||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
| ||p=-2|p=-1|p=0|p=1|p=2|p=3|p=4|
+----++----+----+---+---+---+---+---+
page 4, with differential of degree {-4, 3}:
+----++----+----+---+---+---+---+---+
|q=3 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=2 ||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
| || | | | | 1| | |
|q=1 ||0 |0 |0 |0 |QQ |0 |0 |
+----++----+----+---+---+---+---+---+
| || | | 1| | | | |
|q=0 ||0 |0 |QQ |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=-1||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
|q=-2||0 |0 |0 |0 |0 |0 |0 |
+----++----+----+---+---+---+---+---+
| ||p=-2|p=-1|p=0|p=1|p=2|p=3|p=4|
+----++----+----+---+---+---+---+---+
|