# The Norm Map in (group) cohomology via classifying spaces

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 $\mathcal{N}^G_H$.

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In Benson's book "Representations and Cohomology II" (p. 124 in my edition) there are two references that could be of interest: "Segal [...] has constructed a generalised cohomology theory [...] The transfer map in this theory gives the Evens norm map $H^r(BH;R) \to H^{nr}(BG;R)$ for r even. Fulton and MacPherson [...] give a direct description of the norm map for finite coverings of topological spaces". – Ralph Jan 4 '12 at 0:02

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.

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The article Justin refers to is Localization and completion theorems for $MU$-module spectra''. Annals of Math. 146(1997), 509-544. Its key technical tool is the multiplicative norm map. That is also a key technical tool in the work of Hill, Hopkins, and Ravenel on the Kervaire invariant problem. Their definition exhibits the norm map explicitly on the equivariant spectrum level, Greenlees and May only implicitly. Anna Marie Bohmann has lifted the Greenlees-May construction to the spectrum level and shown that the two coincide. – Peter May Dec 29 '11 at 13:09
Hi Justin and Prof. May! This is nice. Unfortunately, I know nothing about equivariant homotopy and spectrum level stuff; do you have a reference of a good place to start? – Chris Gerig Dec 30 '11 at 13:53
I would also recommend Equivariant homotopy and cohomology theory available off of Peter's website here: math.uchicago.edu/~may/BOOKS/alaska.pdf as well as the course notes of Stefan Schwede which I linked to above. – Justin Noel Dec 30 '11 at 18:32
@Justin: The link is broken, could you fix it / provide another link? And thanks, May's book seems like it will be very informative on top of his other suggestion. – Chris Gerig Jan 3 '12 at 0:51
@Chris: Sorry about that. I'm stilling learning how to use this system. It's fixed now. – Justin Noel Jan 3 '12 at 8:28

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.

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Thanks!... (using up characters to meet the minimum requirement to post this comment) – Chris Gerig Dec 30 '11 at 17:22