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 $res^G_H$ is induced from the map $BH=EG/H\rightarrow EG/G=BG$ (contractible total space $EG$).

**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 $G/H$ (set of left coset representatives). Consider $H$ as a subgroup of $S_H$ through left multiplication.  Then the *wreath product* is the group $S\int H$ defined as $H^{\oplus|G:H|}\rtimes S$, where $s^{-1}(\prod_{x\in G/H}h_x)s=\prod_{x\in G/H}h_{s(x)}$.  We can then define the *monomial representation* $\Phi:G\rightarrow S_{G/H}\int H$ via $\Phi(g)=\pi(g)\prod_{x\in G/H}h_{g,t}$ 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 $\alpha\in H^*(H,k)$ is of even degree, the *norm map* $\mathcal{N}^G_H:H^{even}(H,k)\rightarrow H^*(G,k)$ is defined as $\mathcal{N}^G_H(\alpha)=\Phi^*(1\int \alpha)$.  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 has nice properties that make it 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 (with degrees between $n$ and $n|G:H|$), so that 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 normal in $G$ of prime index $p$ ($\beta$ is any another homogeneous element of even degree).