Nonhomeomorphic conjugate homogeneous spaces

Let X be an algebraic variety over the field of complex numbers C.
Let \sigma be an automorphism of C, not necessarily continuous.
Applying \sigma to the coefficients of the polynomials defining the
algebraic variety X, we obtain a conjugate variety \sigma X. In 1964
J.-P. Serre constructed an example of a smooth projective variety X
and an automorphism \sigma of C such that the topological fundamental
groups of X and of \sigma X are not isomorphic. In the talk I will
describe an example of a *homogeneous space* X=G/H  such that the
topological fundamental groups of X and of \sigma X are not isomorphic
for suitable \sigma. In our example G=SL(5,C) x C^*, and H is any
finite nonabelian subgroup of order 55 of G that projects onto the
subgroup \mu_5 of C^*.  This is a joint work with Yves Cornulier.