The Norm Map in (group) cohomology via classifying spaces

Question

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 $\mathcal{N}^G_H$.

Answer 1

First let me describe the norm in degree 0. Let $k$ be a commutative ring, given $\alpha\in H^0(BH,k)$, which corresponds to a an $H$ equivariant homotopy class of a map $EH_+\rightarrow k$ where $k$ has the trivial $H$ action. W can take the $n$th smash power of this map to obtain \begin{equation*} (EH_+)^{\wedge n}\cong E(H^n)_+\rightarrow k^{\wedge n}. \end{equation*} This map is naturally $\Sigma_n \wr H$ equivariant. Using the (equivariant) multiplication in $k$ we can make this map land in $k$ again. Now use the monomial representation above $G\rightarrow S_{G/H}\wr H\cong \Sigma_n \wr H$, we can regard this $\Sigma_n \wr H$ equivariant map as a $G$ map and $G$ acts freely on the source. Taking orbits we obtain an ordinary map $[BG_+, k]$ as desired.

To move up to higher degrees, I find it helpful to use equivariant spectra. So now our map $\alpha$ in degree $i$ will corresponde to a map $\Sigma^\infty_+ EH \rightarrow S^i \wedge Hk$. We proceed exactly as above but with one little snag. After taking smash powers the $S^i$ becomes $S^{i\rho}$ where $\rho$ is the standard $n$ dimensional representation of $\Sigma_n$ ($H$ acted trivially on $S^i$). When $i$ is even and $k$ is a field of characteristic $p$ (all $i$ in characteristic $2$) we use that $S^{i\rho}$ is $k$-orientable to land us in the correct degree.

For more details on the norm in spectra consult the article by May and Greenlees or Schwede's course notes especially chapter 7.

Answer 2

The easiest starting point for equivariant stable homotopy theory is probably the paper with that title in The Handbook of Algebraic Topology, edited by I.M. James. The paper modern foundations for stable homotopy theory in the same volume is one of several possible starting points for spectrum level stuff.