  We introduce a new technique to prove connectivity of subsets of covering
spaces (so called inductive connectivity), and apply it to Galois theory of
problems of enumerative geometry. As a model example, consider the problem of
permuting the roots of a complex polynomial $f(x) = c_0 + c_1 x^{d_1} + \ldots
+ c_k x^{d_k}$ by varying its coefficients. If the GCD of the exponents is $d$,
then the polynomial admits the change of variable $y=x^d$, and its roots split
into necklaces of length $d$. At best we can expect to permute these necklaces,
i.e. the Galois group of $f$ equals the wreath product of the symmetric group
over $d_k/d$ elements and $\mathbb{Z}/d\mathbb{Z}$. The aim of this paper is to
prove this equality and study its multidimensional generalization: we show that
the Galois group of a general system of polynomial equations equals the
expected wreath product for a large class of systems, but in general this
expected equality fails, making the problem of describing such Galois groups
unexpectedly rich.
