Serre edge homomorphisms and transgression

Let $p:E\to B$ and $R$ satisfy Homological Serre spectral sequence. The fiber-axis Serre edge homomorphism in total degree $n\ge 0$ is ${ϵ}_F:E_{0,n}^2=H_0(B;{\mathcal{H}}_n(p;R))\twoheadrightarrow E_{0,n}^{\mathrm{\infty }}\cong F_0{H}_n(E;R)\hookrightarrow H_n(E;R)\mathrm{.}$ The base-axis Serre edge homomorphism is ${ϵ}_B:H_n(E;R)\twoheadrightarrow H_n(E;R)\mathrm{/}{F}_{n-1}{H}_n(E;R)\cong E_{n,0}^{\mathrm{\infty }}\hookrightarrow E_{n,0}^2=H_n(B;{\mathcal{H}}_0(p;R))\mathrm{.}$ The surjection and inclusion are the finite normalized axis maps of Edge homomorphisms of a first quadrant spectral sequence. In particular, the fiber-axis source is the local-coefficient group of coinvariant type $H_0(B;{\mathcal{H}}_n)$, not an invariant subgroup of one fiber stalk. The base-axis target has coefficients ${\mathcal{H}}_0$ and is not identified with ordinary $H_n(B;R)$ unless a specified coefficient identification permits it.

For $n\ge 2$, no differential before $d_n$ can enter $(n,0)$, while no differential can leave $(0,n-1)$. Consequently the transition maps canonically realize $D_n:=E_{n,0}^n\hookrightarrow E_{n,0}^2,Q_{n-1}:=E_{0,n-1}^n\twoheadleftarrow E_{0,n-1}^2\mathrm{.}$ Thus $D_n$ is exactly the subgroup of base-axis classes surviving $d_2,\dots ,d_{n-1}$, and $Q_{n-1}$ is the fiber-axis group modulo the images of the differentials arriving before page $n$. The homological Serre transgression is the partial homomorphism ${\tau }_n=d_n:D_n⟶Q_{n-1},$ where $d_n$ has source $(n,0)$ and target $(0,n-1)$. This equality fixes the sign: there is no additional sign beyond the convention $d_r:E_{a,b}^r\to E_{a-r,b+r-1}^r$. For $n=2$, the empty list of earlier differentials gives $D_2=E_{2,0}^2$ and $Q_1=E_{0,1}^2$.

Dually, whenever a first-quadrant cohomological Serre spectral sequence with $d_r:E_r^{a,b}\to E_r^{a+r,b-r+1}$ has been supplied, its transgressive fiber classes in degree $n-1$ are the classes in $D^{n-1}:=E_n^{0,n-1}\hookrightarrow E_2^{0,n-1}$ that survive the earlier outgoing differentials. Their cohomological transgression is ${\tau }^n=d_n:D^{n-1}⟶E_n^{n,0},$ whose target is the base-axis quotient by earlier incoming images. This dual clause is conditional on the cohomological sequence; it does not use one as a prerequisite for the homological definition.

These definitions include zero groups, the zero ring, empty axis terms, and classes killed by an earlier differential (which are outside $D_n$ or $D^{n-1}$). They do not define a value for a nonsurviving class, do not select representatives of any quotient, and use no choice principle. Neither definition states a biconditional.