Suppose $p: \tilde X \to X$ is a normal covering space with deck group $\Gamma$. Then there are transfer maps $$ \tau^*: H^*(\tilde X, \mathbb Q) \to H^*(X, \mathbb Q) $$ and $$ \tau_*: H_*(X,\mathbb Q) \to H^*(\tilde X, \mathbb Q). $$ (Suppose I normalize these so that $\tau^* \circ p^* = 1$ and $p_* \circ \tau_* = 1$). There is also the evaluation pairing $$ \langle \cdot, \cdot \rangle: H^k(X, \mathbb Q)\otimes H_k(X, \mathbb Q) \to \mathbb Q $$ and similarly for $\tilde X$. Are the transfer maps then adjoint, in the sense that for all $x \in H_k(X, \mathbb Q)$ and $\alpha \in H^k(\tilde X, \mathbb Q)$, we have $$ \langle\alpha, \tau_*(x)\rangle = \langle\tau^*(\alpha), x\rangle? $$ I've tried consulting Hatcher but he only handles the cohomological transfer map, and Google doesn't turn anything up. It seems reasonable that this should be true, but since I can't find it written down anywhere I figured it couldn't hurt to ask on Stackexchange.
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