C = edgeComplex ESuppose 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.
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The second page of the corresponding spectral sequences take the form:
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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
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To see that it is acyclic we can compute
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The method currently does not support pruned spectral sequences.
The object edgeComplex is a method function.