  In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study
of the space of real univariate polynomials of a given degree d and with no
real roots of multiplicity exceeding a given positive integer. Expanding their
studies, we consider the spaces of real monic univariate polynomials of degree
d whose real divisors avoid sequences of root multiplicities taken from a given
poset of compositions which is closed under certain natural combinatorial
operations. In this paper, we concentrate on the fundamental group of such
spaces. We find explicit presentations for the fundamental groups in terms of
generators and relations and show that in a number of cases they are free with
rank bounded from above by a quadratic function in d. We also show that the
fundamental group stabilizes for d large. We further show that the fundamental
groups admit an interpretation as special bordisms of immersions of 1-manifolds
into the cylinder S^1 \times R, whose images avoid the tangency patterns from
the poset with respect to the generators of the cylinder.
