Characterization of the transfer map in group theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T16:50:19Z http://mathoverflow.net/feeds/question/83790 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83790/characterization-of-the-transfer-map-in-group-theory Characterization of the transfer map in group theory Martin Brandenburg 2011-12-18T12:39:18Z 2011-12-21T08:23:06Z <p>Let $i : H \to G$ be a subgroup of finite index. The <a href="http://en.wikipedia.org/wiki/Transfer_%28group_theory%29" rel="nofollow">transfer map</a> is a special homomorphism $V(i) : G^\mathrm{ab} \to H^\mathrm{ab}$. The usual ad hoc definition uses a set of representatives of $H$ in $G$ and then you have to check that it is independent from this choice and that it is a homomorphism at all. I think this definition is not enlightening at all (although it is, of course, useful for explicit calculations). A better one uses group homology. Namely, for a $G$-module $A$ there is a natural transformation <code>$A_G \to \mathrm{res}^{G}_{H} A_H$</code>, $[a] \mapsto \sum_{Hg \in H/G} [ga]$, which extends to a natural transformation $H_*(G;A) \to H_*(H;\mathrm{res}^{G}_{H} A)$ (usually called corestriction or transfer). Now evaluate at $A = \mathbb{Z}$ and $* = 1$ to get $G^\mathrm{ab} \to H^\mathrm{ab}$. One can then calculate this map using the explicit isomorphisms and homotopy equivalences involved; but now you know by the general theory that it is a well-defined homomorphism.</p> <p>It also follows directly that the transfer is actually a functor <code>$V : \mathrm{Grp}_{mf} \to \mathrm{Ab}^{\mathrm{op}}$</code> with object function $G \mapsto G^{\mathrm{ab}}$, where $\mathrm{Grp}_{mf}$ is the category whose objects are groups and whose morphisms are monomorphisms of finite index.</p> <p>I would like to know if there is an even more "abstract" definition. To be more precise: Is there a categorical characterization of the functor $V$ which only uses the adjunction $\mathrm{Grp} {\longleftarrow \atop \longrightarrow} \mathrm{Ab}$?</p> <p>Edit: There are many interesting answers so far which give, in fact, very "enlightening" definitions of the transfer. But I would also like to know if there is a pure categorical one, such as the one given by Ralph.</p> <p>Edit: A very interesting note by Daniel Ferrand is <a href="http://people.math.jussieu.fr/~dferrand/Transfer.pdf" rel="nofollow">A note on transfer</a>. There a more general statement is proven (even in a topos setting): Let $G$ act freely on a set $X$ such that $X/G$ is finite with at least two elements. Then there is an <em>isomorphism</em> of abelian groups $(\mathrm{Ver},\mathrm{sgn}) : {\mathrm{Aut}_{G}(X)}^{\mathrm{ab}} \cong G^{\mathrm{ab}} \times \mathbb{Z}/2$. It is natural with respect to $G$-isomorphisms. Here again I would like to ask if it is possible to characterize this isomorphism by its properties (instead of writing it down via choices, whose independence has to be shown afterwards).</p> <p>Proposition 7.1. in this paper includes the interpretation via determinants mentioned by Geoff in his answer, actually something more general: For w.l.o.g. abelian $G$ there is a commutative diagram</p> <p>$\begin{matrix} {\mathrm{Aut}_{G}(X)}^{\mathrm{ab}} & \cong & \mathrm{Aut}_{\mathbb{Z}G}{\mathbb{Z}X}^{\mathrm{ab}} \\\\ \downarrow & & \downarrow \\\\ G \times \mathbb{Z}/2 & \rightarrow & (\mathbb{Z} G)^{x} \end{matrix}$</p> <p>Thus we may think of transfer and signature as the embedding the standard units into the group ring.</p> http://mathoverflow.net/questions/83790/characterization-of-the-transfer-map-in-group-theory/83791#83791 Answer by Geoff Robinson for Characterization of the transfer map in group theory Geoff Robinson 2011-12-18T13:45:13Z 2011-12-18T15:56:40Z <p>It is not really categorical, so this is maybe more of a comment than an answer, but the way I find easiest to see that transfer really gives a homomorphism (independent of choice of coset representatives, but it's not clear to me that this issue is much easier from this viewpoint) is from a viewpoint which may be due to T. Yoshida, who wrote some papers on "character-theoretic transfer" in the 70s. Given that $[G:H]$ is finite, consider a group homomorphism $\phi: H \to A$ where $A$ is an Abelian group. Let $R$ be the group ring ${\rm GF}(2)[A].$ Consider $\phi$ as a rank $1$-representation of $H$ over $R$. Induce that to a representation from $G \to {\rm GL}_d(R),$ where $d = [G:H]$, and take the determinant of that induced representation. In the case that <code>$A = H/H^{\prime}$</code>, we (implicitly) obtain the homomorphism <code>$V_G: G^{ab} \to H^{ab}.$</code>.</p> http://mathoverflow.net/questions/83790/characterization-of-the-transfer-map-in-group-theory/83794#83794 Answer by Benjamin Steinberg for Characterization of the transfer map in group theory Benjamin Steinberg 2011-12-18T15:06:06Z 2011-12-18T15:06:06Z <p>An alternative description along Geoff's line is the following. Let $T$ be a set of coset reps for $H$. Then associated to $T$ is a Krasner-Kaloujnine embedding $$G\hookrightarrow H^{G/H}\rtimes (G/H_G)$$ where $H_G$ is the intersection of the conjugates of $H$. This embedding depends on $T$ only up to an inner automorphism of $H^{G/H}$. The abelianization of the semidirect product above is $H^{ab}\times (G/H_G)^{ab}$ and the restriction of the abelianization map to $G$ yields a homomorphism $$G\hookrightarrow H^{ab}\times (G/H_G)^{ab}\to H^{ab}$$ where the last map is the projection. This induces a homomorphism $G^{ab}\to H^{ab}$ which is the transfer. The independence from $T$ follows the independence of the embedding up to inner automorphism. </p> http://mathoverflow.net/questions/83790/characterization-of-the-transfer-map-in-group-theory/83823#83823 Answer by Andy Putman for Characterization of the transfer map in group theory Andy Putman 2011-12-18T20:24:31Z 2011-12-18T20:24:31Z <p>My answer is also not categorical, but it is too long for a comment and I think it sheds light on the nature of the transfer.</p> <p>I think of the transfer as really being a fact about covering spaces. Let $\pi : X \rightarrow Y$ be a degree $n$ covering map. If $\sigma : \Delta^k \rightarrow Y$ is a singular $k$-simplex on $Y$, then covering space theory provides $n$ different lifts $\tilde{\sigma}_1,\ldots,\tilde{\sigma}_n : \Delta^k \rightarrow X$ of $\sigma$. Define $\tau_k(\sigma)$ to be the singular $k$-chain $\tilde{\sigma}_1 + \cdots+ \tilde{\sigma}_n$ on $X$. This extends by linearity to a map $\tau_k : C_k(Y;R) \rightarrow C_k(X;R)$, where $R$ is any commutative ring and $C_{\ast}(\cdot,R)$ is the abelian group of singular simplices with coefficients in $R$. It is clear that the $\tau_k$ combine together to form a chain map $\tau : C_{\ast}(Y;R) \rightarrow C_{\ast}(X;R)$ that satisfies $$\pi_{\ast}(\tau(x)) = n \cdot x,$$ where $\pi_{\ast} : C_{\ast}(X;R) \rightarrow C_{\ast}(Y;R)$ is the map on singular chains induced by $\pi$. The transfer map $H_{\ast}(Y;R) \rightarrow H_{\ast}(X;R)$ is the map on homology induced by $\tau$.</p> <p>To recover the classical transfer, let $Y$ be a $K(G,1)$ and $X$ be the cover corresponding to $H$. </p> http://mathoverflow.net/questions/83790/characterization-of-the-transfer-map-in-group-theory/83824#83824 Answer by Benjamin Steinberg for Characterization of the transfer map in group theory Benjamin Steinberg 2011-12-18T21:12:44Z 2011-12-18T21:12:44Z <p>Here is another answer. It is in fact equivalent to all the previous answers but is more categorical. Let $X$ be a finite transitive $G$-set and let $\mathcal G=G\ltimes X$ be the corresponding Grothendieck construction. So it is the groupoid with objects $X$ and arrows $(g,x):x\to gx$. The product is $(g,hx)(h,x)=(gh,x)$. It is the groupoid analogue of the covering space of $BG$ associated to the $G$-set $X$. </p> <p>Now if $H$ is an isotropy group, then $\mathcal G$ is equivalent to $H$ but the choice of equivalence is not unique. This is Martin's complaint. But since any two naturally equivalent functors from a groupoid to an abelian group are the same, there is a CANONICAL functor $\tau\colon \mathcal G\to H^{ab}$ which is just the universal functor from $\mathcal G$ to an abelian group. </p> <p>The tranfer is the map $$g\mapsto \sum_{x\in X}\tau(g,x).$$</p> http://mathoverflow.net/questions/83790/characterization-of-the-transfer-map-in-group-theory/83843#83843 Answer by Ralph for Characterization of the transfer map in group theory Ralph 2011-12-19T02:08:13Z 2011-12-20T06:53:43Z <p><strong>Edit:</strong> As Martin remarked, there is a gap in the proof below. It can be closed by replacing axiom 1 by 1'. However, this isn't very satisfying, as it lowers the categorial flauvor of the characterization. Perhaps one should further investigate, if axiom 1 couldn't be used anyway. </p> <p>$\hspace{5pt}$1'. If $G=\langle H,x \rangle, n=(G:H)$ and $h \in \cap_{i=0}^{n-1}x^iHx^{-i}$, then $t^G_H(h[G,G])$ is represented $\hspace{10pt}$ $\hspace{12pt}$ by $(hx)^nx^{-n}$. </p> <p>Futhermore axiom 3 should be </p> <p>$\hspace{5pt}$3. If $f: G \to G'$ is a homomorphism, $H' \le G, H = f^{-1}(H')$ and $(G:H) = (G':H'),$ $\hspace{5pt}$ $\hspace{12pt}$then the diagram commutes. </p> <hr> <p>As far as I can see, the answers above are all concerned with an explicit construction of the transfer. Here I will go the other direction and characterize the transfer by its properties. Let $V$ denote the usual transfer. </p> <blockquote> <p>Suppose for each pair $H \le G$ with $(G:H) &lt; \infty$ there is a homomorphism $t^G_H: G_{ab} \to H_{ab}$ satisfying the subsequent properties. Then $t^G_H = V^G_H$. </p> </blockquote> <ol> <li><p>The composition $G_{ab}\hspace{1pt} \xrightarrow{ t } \hspace{1pt} H_{ab} \hspace{1pt} \xrightarrow{\bar{i}} \hspace{1pt} G_{ab}$ is multiplication by $(G:H)$. </p></li> <li><p>If $H \le K \le G$ then $t^K_H \circ t^G_K = t^G_H$ </p></li> <li><p>If $(G:H) = (G':H')$ and $f: G \to G'$ is a homomorphism with $f(H) \le H'$ then the following diagram commutes: $$\begin{array}{ccc} G_{ab} &amp; \xrightarrow{\bar{f}} &amp; G_{ab}' \newline t \downarrow &amp; &amp; \downarrow t' \newline H_{ab} &amp; \xrightarrow[\bar{f}]{} &amp; H_{ab}' \end{array}$$</p></li> </ol> <hr> <p><em>Proof:</em> a) It's well-known that $V$ satisfies $1.-3.$. </p> <p>b) By 1., $t^G_H$ and $V^G_H$ agree on $\bar{x}$ for $x \in H$. </p> <p>c) Suppose $G = \langle H, x \rangle$ and $(G:H) = n$. Let $f: \mathbb{Z} \to G, 1 \mapsto x$. By 1. we have $t: \mathbb{Z} \to n\mathbb{Z}, 1 \to n$. Now 3. implies $t^G_H(\bar{x}) = \bar{x}^n = V^G_H(\bar{x})$. In particular $t^G_G = id|G_{ab} = V^G_G$. </p> <p>d) We show by induction on $n=(G:H)$ that $t^G_H = V^G_H$ for all $H \le G$. The case $n=1$ was shown in c). Suppose $n>1$ and $t^G_H = V^G_H$ holds for all $H \le G$ with $(G:H) &lt; n$. Let $x \in G$. If $G = \langle H, x \rangle$ then $t^G_H(\bar{x}) = V^G_H(\bar{x})$ by c). So assume $K := \langle H, x \rangle$ is a proper subgroup of $G$. Because of b) we may assume $x \notin H$. Thus $(G:K),(K:H) &lt; n$ and we conclude from 2. and the induction hypothesis and a) that $t^G_H = V^G_H$. <em>q.e.d.</em> </p> http://mathoverflow.net/questions/83790/characterization-of-the-transfer-map-in-group-theory/83947#83947 Answer by Ronnie Brown for Characterization of the transfer map in group theory Ronnie Brown 2011-12-20T14:37:13Z 2011-12-20T14:37:13Z <p>It might be useful here to make the translation between the arguments using covering spaces and those using the Grothendieck construction by noting that there is a well known equivalence for a groupoid $G$ between the category of actions of $G$ on sets and that of <em>covering morphisms</em> of the groupoid $G$. See for example </p> <p>Higgins, P.J., Notes on categories and groupoids, Mathematical Studies, Volume 32. Van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1-195. (downloadable)</p> <p>I have traced this notion of covering morphism back to a 1951 Annals. of Math. paper by P.A. Smith (under the name <em>regular morphism</em>), and the equivalence mentioned above is of course related to the so-called Grothendieck construction, though is was earlier considered by C. Ehresmann. </p> <p>I believe there are advantages in an exposition of covering spaces using this notion, since a covering <em>map</em> is nicely modelled by a covering <em>morphism</em>. </p>