The well-known *transfer map* in group (co)homology can be defined with only homological algebra, or with algebraic topology via classifying spaces (group cohomology of $G$ is isomorphic to ordinary cohomology of $BG$). Here I fix a group $G$ with finite-index subgroup $H$, and field $k=\mathbb{Z}_p$. Let $p:\tilde{Y}\rightarrow Y$ be a covering map, where $Y$ is a $K(G,1)$-manifold of top dimension $n$ and $\tilde{Y}$ is the cover which corresponds to $H$ (so that it is a $K(H,1)$-manifold); clearly $p$ is a $|G:H|$-sheeted covering map. Now the transfer map $tr^G_H:H_n(G,k)\rightarrow H_n(H,k)$ agrees with the transfer map $H_n(Y;k)\rightarrow H_n(\tilde{Y};k)$, which in terms of chain complexes is induced by $\sigma\mapsto\sum_{\sigma'}\sigma'$ where $\sigma$ is an oriented $n$-cell of $Y$ and $\sigma'$ ranges over the oriented $n$-cells of $\tilde{Y}$ lying over $\sigma$. In cohomology this is $f\mapsto[\sigma\mapsto \sum_{\sigma'}f(\sigma')]$ for cochains. Likewise, the *restriction map* runs in the opposite direction of the transfer and is induced from the map $BH=EG/H\rightarrow EG/G=BG$.

**What is the corresponding classifying space construction in the setting of the** *norm map* **defined below?**

Let $S_{G/H}$ be the group of permutations of $G/H$ (left coset representatives) and consider $H$ as a subgroup of $S_H$ through left multiplication. The *wreath product* $S_{G/H}\int H$ is $H^{\oplus|G:H|}\rtimes S_{G/H}$, where $s^{-1}(\prod_{x\in G/H}h_x)s=\prod_{x\in G/H}h_{s(x)}$. From this is the *monomial representation* $\Phi:G\rightarrow S_{G/H}\int H$ defined by $\Phi(g)=\pi(g)\prod_{x\in G/H}h_{g,t}$ where $gx=x_gh_{g,t}$ (for $x,x_g\in G/H$ and $h_{g,t}\in H$), and $x\mapsto x_g$ induces a permutation $\pi(g)\in S_{G/H}$ (so $\pi$ is the representation of $G$ as a group of permutations of its left coset space $G/H$).

Finally, for $\alpha\in H^*(H,k)$ of even degree, the *norm map* $\mathcal{N}^G_H:H^{even}(H,k)\rightarrow H^*(G,k)$ is defined by $\mathcal{N}^G_H(\alpha)=\Phi^*(1\int \alpha)$; for convenience I left out the cohomological construction of the element $1\int\alpha$ from $\alpha$. If $\alpha\in H^n(H,k)$ then $\mathcal{N}^G_H(\alpha)\in H^{n|G:H|}(G,k)$. This is a pretty hard construction for me to grasp, but ultimately it is extremely useful in group cohomology (in particular, it was defined by Leonard Evens and used to prove the finite generation result that $H^*(G,\mathbb{Z}_p)$ is Noetherian for any $p$ dividing $|G|$).

The reason this question arose is because $\mathcal{N}^G_H(1+\alpha)=1+tr^G_H(\alpha)+\cdots+\mathcal{N}^G_H(\alpha)$ for $\alpha\in H^n(H,k)$, where the intermediate terms are also transfers, i.e. the norm map is intertwined with the transfer map. Even simpler, $\mathcal{N}^G_H(\alpha+\beta)=\mathcal{N}^G_H(\alpha)+tr^G_H(\mu)+\mathcal{N}^G_H(\beta)$ for some $\mu\in H^*(H,k)$, if $H$ is an index-$p$ normal subgroup ($\alpha,\beta$ are homogeneous elements of even degree).

[[Addendum]] I have actually just stumbled upon a piece of this desired construction, on pg73-75 of Adem & Milgram's *Cohomology of Finite Groups*. The map given here is $BG\rightarrow (BH)^{|G:H|}\times_{S_{G/H}}ES_{G/H}\simeq B(S_{G/H}\int H)$ and is induced from $\Phi$. I assume from here we can functorially relate cohomology classes of $H$ to that of $S_{G/H}\int H$ and hence obtain $\mathcal{N}^G_H$.