I am interested in whether the transgression maps for group cohomology and group homology are related via the universal coefficient theorem.

Let $G$ be a group, $H$ a normal subgroup of $G$ and let $A$ and $B$ be $G$-modules. The Lyndon-Hochschild-Serre spectral sequences in cohomology and homology give transgression maps:

$$d^2 : H^1(H,A)^{G/H} \rightarrow H^2(G/H, A^H),$$

$$d_2 : H_2(G/H, B_H) \rightarrow H_1(H, B)_{G/H}.$$

Via the universal coefficients theorem, applied functorially to $d^2$, we get a commutative diagram

\begin{array}{cccc} H^1(H,A)^{G/H} &\xrightarrow{d^2}&H^2(G/H, A^H)\\ \downarrow & & \downarrow \\ \operatorname{Hom}(H_1(H;\mathbb{Z}),A)^{G/H} & \xrightarrow{F} & \operatorname{Hom}(H_2(G/H;\mathbb{Z}),A^H) \end{array}

Suppose that $A$ is torsion free and $B = A^\vee := \operatorname{Hom}(A, \mathbb{C}^\times)$.

• Is $F$ equal to $$d_2^{\vee} : \operatorname{Hom}(H_1(H,A^{\vee})_{G/H}, \mathbb{C}^{\times}) \rightarrow \operatorname{Hom}(H_2(G/H, (A^{\vee})_H), \mathbb{C}^{\times})?$$
• Does anyone know a reference for such a statement, and whether it generalizes to the case that $A$ is not torsion-free (used in identifying $H_1(H, A^\vee)$ with $H_1(H, \mathbb{Z}) \otimes A^\vee$)?