Naturality of the transfer in group cohomology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T06:15:15Z http://mathoverflow.net/feeds/question/119470 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119470/naturality-of-the-transfer-in-group-cohomology Naturality of the transfer in group cohomology Mark Grant 2013-01-21T14:03:24Z 2013-01-21T14:25:32Z <p>Let $G$ be a (discrete) group and $H\le G$ a subgroup of finite index. Then there is a transfer map $$tr\colon\thinspace H^\ast(H;M)\to H^\ast(G;M)$$ in group cohomology, where $M$ is any $G$-module (see Brown's "Cohomology of groups", Chapter III).</p> <p>I think this construction should be natural, in the following sense. Let $f\colon\thinspace G'\to G$ be a homomorphism such that $H':=f^{-1}(H)$ is of finite index in $G'$, and let $M$ be a $G$-module. Then the following diagram commutes, where the horizontal maps are transfers: $$\begin{array}{ccc} H^\ast(H;M) &amp; \to &amp; H^\ast(G;M) \newline \downarrow f^\ast &amp; &amp; \downarrow f^\ast \newline H^\ast(H';f^\ast M) &amp; \to &amp; H^\ast(G';f^\ast M) \end{array}$$ Note that I do not want to assume that $(G':H')=(G:H)$ (however, I am willing to assume that $H\le G$ and $H'\le G'$ are normal, if necessary).</p> <blockquote> <p>Does anyone know of a reference for this naturality?</p> </blockquote> http://mathoverflow.net/questions/119470/naturality-of-the-transfer-in-group-cohomology/119473#119473 Answer by Oscar Randal-Williams for Naturality of the transfer in group cohomology Oscar Randal-Williams 2013-01-21T14:25:32Z 2013-01-21T14:25:32Z <p>I don't believe this is true. Let $(G, H) = (\Sigma_3, C_3)$ and $f : C_3 \to \Sigma_3$. Then your square says that $$H^1(C_3;\mathbb{Z}/3) = \mathbb{Z}/3 \longrightarrow H^1(\Sigma_3;\mathbb{Z}/3) = 0\longrightarrow H^1(C_3;\mathbb{Z}/3) = \mathbb{Z}/3$$ is the identity, which is false.</p>