On what kind of objects do the Galois groups act? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:58:38Z http://mathoverflow.net/feeds/question/69617 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69617/on-what-kind-of-objects-do-the-galois-groups-act On what kind of objects do the Galois groups act? semyon alesker 2011-07-06T09:25:30Z 2011-12-20T10:35:54Z <p>I am neither number theorist nor algebraic geometer. I am wondering whether Galois groups of number fields (say the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which are not related a priori to the number theory.</p> <p>I am aware of two such situations of rather different nature:</p> <p>(1) Grothendieck's dessins d'enfants: $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on certain graphs on 2-dimensional surfaces.</p> <p>(2) $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the profinite completion of the topological $K$-theory (of sufficiently nice spaces, e.g. finite $CW$-complexes).</p> <p>As far as I understand (am I wrong?) the most important and best studied examples of actions of Galois groups are actions on $l$-adic cohomology of varieties over number fields. But this is not what I am looking for: number fields appear in the formulation of the problem from the vary beginning.</p> http://mathoverflow.net/questions/69617/on-what-kind-of-objects-do-the-galois-groups-act/69628#69628 Answer by SGP for On what kind of objects do the Galois groups act? SGP 2011-07-06T11:57:41Z 2011-07-06T11:57:41Z <p>The work of Dennis Sullivan (see <a href="http://mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0169.0176.ocr.pdf" rel="nofollow">his ICM 1970 address</a> for a summary) is a veritable source of such unexpected Galois actions.</p> http://mathoverflow.net/questions/69617/on-what-kind-of-objects-do-the-galois-groups-act/69633#69633 Answer by Adrien for On what kind of objects do the Galois groups act? Adrien 2011-07-06T13:04:52Z 2011-07-08T20:36:04Z <p>There are some nice examples in knot theory and quantum algebra.</p> <p>If $X$ is an algebraic variety over $\mathbb{Q}$, there is a canonical outer action $G_{\mathbb{Q}} \rightarrow Out(\hat{\pi}_1(X(\mathbb{C})))$ (which also exists for other fields than $\mathbb{Q}$ using the so-called algebraic fundamental group but I don't want to say something wrong about that). Roughly speaking, this is because finite covering of $X$ can be defined over $\bar{\mathbb{Q}}$, together with the relation between (regular) finite covering and finite quotients of the fundamental group. This action thus has the same origine as for dessin d'enfants. </p> <p>Of particular interest is the study of this action in the case $X$ is the moduli space of algebraic curves of genus $g$ with $n$ marked points. This was suggested in Grothendieck's esquisse, and leads to the so-called Grothendieck-Teichmuller theory which gives a rather explicit description of a group which actually contains $G_{\mathbb{Q}}$. (see <a href="http://mathoverflow.net/questions/2138/cartographic-group-and-flat-stringy-connection/63748#63748" rel="nofollow">http://mathoverflow.net/questions/2138/cartographic-group-and-flat-stringy-connection/63748#63748</a> or <a href="http://mathoverflow.net/questions/64065/where-is-a-good-place-to-start-learning-about-the-grothendieck-teichmuller-group" rel="nofollow">http://mathoverflow.net/questions/64065/where-is-a-good-place-to-start-learning-about-the-grothendieck-teichmuller-group</a>)</p> <p>Actually, there are several flavours of the Grothendieck-Teichmuller group: a profinite one $\widehat{GT}$ which does contain $G_{\mathbb{Q}}$ and a group $GT(k)$ defined for every field $k$. A deep result of Drinfeld assert that this latter group is in some sense a universal automorphism group of braided monoidal categories. Indeed, it acts on the set of Drinfeld associator with coefficients in $k$.</p> <p>Now, there is also a morphism </p> <p>$G_{\mathbb{Q}}\rightarrow GT(\mathbb{Q}_{\ell})$</p> <p>for every prime number $\ell$. Hence the absolute galois group acts on each kind of object in which associators come up (assuming that it is in a situation where one can work over $\mathbb{Q}_{\ell}$). It leads to, I think, quite surprizing examples like action on finite type invariants of knots and links, on quantization functor of Lie bialgebras, and several other constructions arising in deformation/quantization theory as they are often related to Drinfeld associators. </p> http://mathoverflow.net/questions/69617/on-what-kind-of-objects-do-the-galois-groups-act/69636#69636 Answer by Cam McLeman for On what kind of objects do the Galois groups act? Cam McLeman 2011-07-06T14:14:09Z 2011-07-06T14:14:09Z <p>This is not exactly an incarnation of the question you asked, in the sense that is not so much an action of a Galois group but rather an action whose existence is governed by a Galois group of number-theoretic origin, but it seems likely to be of interest.</p> <p>Let $K$ be a number field, and let $K^{(1)}$ be the maximal unramified abelian extension of $K$. The Galois group of $K^{(1)}/K$ is a subquotient of Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$) which is isomorphic to the class group of $K$. Note that by Minhyong Kim's answer <a href="http://mathoverflow.net/questions/49960/are-class-numbers-encoded-in-the-absolute-galois-group-of-mathbb-q/49967#49967" rel="nofollow">here</a>, we can characterize this subquotient purely Galois-theoretically. Several authors have discovered surprising links between the arithmetic of number fields and actions of groups on spheres. In particular, when $K$ is the real cyclotomic field $K_m=\mathbb{Q}(\zeta_m+\zeta_m^{-1})$, the class group appears to govern the free actions of binary dihedral groups on spheres $S^n$ with $n\equiv 3\pmod{4}$. Let me loosely quote/paraphrase from Lang's "Units and Class Groups in Number Theory and Algebraic Geometry" (bolding mine):</p> <blockquote> C. T. C. Wall has already shown to depend in part on the 2-primary component of the ideal class group in real cyclotomic fields $K_m^+$ for suitable $m$...Using the algebraic background of a paper of Wall, applied to the surgery exact sequence, Thomas gives examples for the binary dihedral group $D_{4p}$ of order $4p$ operating freely on $S^{4k-1}$ with $k\geq 2$, <b>when the order of</b> $[(K_p^+)^{( 1)}:K_p^+]$ <b>is odd.</b> <br> <p>... <br> Furthermore, according to Thomas, there exist free actions by $D_{4p}$ which can be topologically distinguished <b>only</b> by an invariant in the 2-primary part of the ideal class group of $K_p^+$. </blockquote> <p>Perhaps needless to say, the study of these degrees $[(K_p^+)^{( 1)}:K_p^+]$, even their 2-part, is of tremendous interest in algebraic number theory (Vandiver's conjecture, etc.), so the link to actions on spheres is surprising.</p>