# Edge homomorphisms

Suppose that $E$ is a spectral sequence with the properties that:

1. $E^2_{p,q} = 0$ for all $p < l$ and all $q \in \mathbb{Z}$;

2. $E^2_{p,q} = 0$ for all $q < m$ and all $p \in \mathbb{Z}$;

3. $E$ converges to the graded module $\{H_n\}$ for $n \in \mathbb{Z}$.

Then $E$ determines a $5$-term exact sequence $H_{l+m+2} \rightarrow E^2_{l+2,m} \rightarrow E^2_{l,m+1} \rightarrow H_{l+m+1} \rightarrow E^2_{l+1,m} \rightarrow 0$ which we refer to as the edge complex.

Note that the above properties are satisfied if $E$ is the spectral sequence determined by a bounded filtration of a bounded chain complex.

The following is an easy example, of a spectral sequence which arises from a nested chain of simplicial complexes, which illustrates this concept.

 i1 : A = QQ[a,b,c,d]; i2 : D = simplicialComplex {a*d*c, a*b, a*c, b*c}; i3 : F2D = D; i4 : F1D = simplicialComplex {a*c, d}; i5 : F0D = simplicialComplex {a,d}; i6 : K = filteredComplex({F2D, F1D, F0D},ReducedHomology => false); i7 : C = K_infinity; i8 : prune HH C o8 = -1 : 0 1 0 : QQ 1 1 : QQ 2 : 0 o8 : GradedModule

The second page of the corresponding spectral sequences take the form:

 i9 : E = spectralSequence(K); i10 : e = prune E; i11 : E^2 +------------------------------------------------+------------------------------------------------------------------+---------------------------------------+ o11 = |subquotient (| 1 0 1 0 0 0 0 |, | 1 0 0 0 0 |)|subquotient (| 0 |, | 0 |) |image 0 | | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 | | | -1 | | -1 | | | | | 0 0 -1 0 0 0 0 | | -1 0 0 0 0 | | | 1 | | 1 | |{2, 0} | | | 0 1 0 0 0 0 0 | | 0 0 0 0 0 | | | 0 | | 0 | | | | | | -1 | | -1 | | | |{0, 0} | | | | |{1, 0} | | +------------------------------------------------+------------------------------------------------------------------+---------------------------------------+ |image 0 |subquotient (| 1 0 0 1 1 1 0 0 1 0 |, | 1 1 1 0 0 1 0 |)|subquotient (| 0 1 0 0 0 |, | 0 0 |)| | | | 0 0 0 -1 0 0 1 0 0 0 | | -1 0 0 1 0 0 0 | | | 0 -1 1 -1 1 | | -1 1 | | |{0, -1} | | 0 1 0 0 -1 0 -1 1 0 0 | | 0 -1 0 -1 1 0 0 | | | 1 0 0 1 0 | | 1 0 | | | | | 0 0 1 0 0 -1 0 -1 0 1 | | 0 0 -1 0 -1 0 1 | | | 0 1 0 0 0 | | 0 0 | | | | | | 0 0 1 -1 0 | | -1 0 | | | |{1, -1} | | | | |{2, -1} | +------------------------------------------------+------------------------------------------------------------------+---------------------------------------+ o11 : SpectralSequencePage i12 : e^2 +-------+-------+-------+ | 2 | | | o12 = |QQ |0 |0 | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | | 2 | |0 |0 |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ o12 : SpectralSequencePage

The acyclic edge complex for this example has the form $H_1(C) \rightarrow E^2_{2,-1} \rightarrow E^2_{0,0} \rightarrow H_0(C) \rightarrow E^2_{1, -1} \rightarrow 0$ and is given by

 i13 : edgeComplex E o13 = subquotient (| 1 0 0 1 1 1 0 0 1 0 |, | 1 1 1 0 0 1 0 |) <-- cokernel | 1 1 1 0 0 | <-- subquotient (| 1 0 1 0 0 0 0 |, | 1 0 0 0 0 |) <-- subquotient (| 0 1 0 0 0 |, | 0 0 |) <-- subquotient (| 1 0 |, | 0 |) | 0 0 0 -1 0 0 1 0 0 0 | | -1 0 0 1 0 0 0 | | -1 0 0 1 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 | | 0 -1 1 -1 1 | | -1 1 | | -1 1 | | -1 | | 0 1 0 0 -1 0 -1 1 0 0 | | 0 -1 0 -1 1 0 0 | | 0 -1 0 -1 1 | | 0 0 -1 0 0 0 0 | | -1 0 0 0 0 | | 1 0 0 1 0 | | 1 0 | | 0 -1 | | 1 | | 0 0 1 0 0 -1 0 -1 0 1 | | 0 0 -1 0 -1 0 1 | | 0 0 -1 0 -1 | | 0 1 0 0 0 0 0 | | 0 0 0 0 0 | | 0 1 0 0 0 | | 0 0 | | 1 0 | | 0 | | 0 0 1 -1 0 | | -1 0 | | 0 1 | | -1 | 0 1 2 3 4 o13 : ChainComplex i14 : prune edgeComplex E 1 2 2 1 o14 = 0 <-- QQ <-- QQ <-- QQ <-- QQ 0 1 2 3 4 o14 : ChainComplex

To see that it is acyclic we can compute

 i15 : prune HH edgeComplex E o15 = 0 : 0 1 : 0 2 : 0 3 : 0 4 : 0 o15 : GradedModule

## Caveat

The method currently does not support pruned spectral sequences.