TITLE: On what kind of objects do the Galois groups act?
QUESTION [40 upvotes]: 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.
I am aware of two such situations of rather different nature:
(1) Grothendieck's dessins d'enfants:
$Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on certain graphs (with extra properties and extra data) on 2-dimensional surfaces.
(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).
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.

REPLY [7 votes]: 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.
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 here, 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):

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 $k\geq 2$, when the order of $[(K_p^+)^{( 1)}:K_p^+]$ is odd.

...

Furthermore, according to Thomas, there exist free actions by $D_{4p}$ which
can be topologically distinguished only by an invariant in the 2-primary part
of the ideal class group of $K_p^+$.

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.