Editing Spectral sequence (section)

== Edge maps and transgressions ==

=== Homological spectral sequences ===
Let <math>E^r_{p, q}</math> be a spectral sequence. If <math>E^r_{p, q} = 0</math> for every ''q'' < 0, then it must be that: for ''r'' ≥ 2,
:<math>E^{r+1}_{p, 0} = \operatorname{ker}(d: E^r_{p, 0} \to E^r_{p-r, r-1})</math>
as the denominator is zero. Hence, there is a sequence of monomorphisms:
:<math>E^{r}_{p, 0} \to E^{r-1}_{p, 0} \to \dots \to E^3_{p, 0} \to E^2_{p, 0}</math>.
They are called the edge maps. Similarly, if <math>E^r_{p, q} = 0</math> for every ''p'' < 0, then there is a sequence of epimorphisms (also called the edge maps):
:<math>E^2_{0, q} \to E^3_{0, q} \to \dots \to E^{r-1}_{0, q} \to E^r_{0, q}</math>.
The [[transgression map|transgression]] is a partially-defined map (more precisely, a [[additive relation|map from a subobject to a quotient]])
:<math>\tau: E^2_{p, 0} \to E^2_{0, p - 1}</math>
given as a composition <math>E^2_{p, 0} \to E^p_{p, 0} \overset{d}\to E^p_{0, p-1} \to E^2_{0, p - 1}</math>, the first and last maps being the inverses of the edge maps.{{sfn|May|loc=§ 1}}

=== Cohomological spectral sequences ===
For a spectral sequence <math>E_r^{p, q}</math> of cohomological type, the analogous statements hold. If <math>E_r^{p, q} = 0</math> for every ''q'' < 0, then there is a sequence of epimorphisms
:<math>E_{2}^{p, 0} \to E_{3}^{p, 0} \to \dots \to E_{r-1}^{p, 0} \to E_r^{p, 0}</math>.
And if <math>E_r^{p, q} = 0</math> for every ''p'' < 0, then there is a sequence of monomorphisms:
:<math>E_{r}^{0, q} \to E_{r-1}^{0, q} \to \dots \to E_{3}^{0, q} \to E_2^{0, q}</math>.
The transgression is a not necessarily well-defined map:
:<math>\tau: E_2^{0, q-1} \to E_2^{q, 0}</math>
induced by <math>d: E_q^{0, q-1} \to E_q^{q, 0}</math>.

=== Application ===
Determining these maps are fundamental for computing many differentials in the [[Serre spectral sequence]]. For instance the transgression map determines the differential{{sfn|Hatcher|pp=540, 564}}
:<math>d_n:E_{n,0}^n \to E_{0,n-1}^n</math>
for the homological spectral spectral sequence, hence on the Serre spectral sequence for a fibration <math>F \to E \to B</math> gives the map
:<math>d_n:H_n(B) \to H_{n-1}(F)</math>.