Real polynomials with constrained real divisors. I. Fundamental groups

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 P-d(c Theta) of real monic univariate polynomials of degree d whose real divisors avoid sequences of root multiplicities, taken from a given poset Theta Theta of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of P-d(c Theta) and of some related topological spaces. We find explicit presentations for the groups pi(1)(P-d(c Theta)) 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 pi(1)(P-d(c Theta)) stabilizes for d large. The mechanism that generates pi(1)(P-d(c Theta)) has similarities with the presentation of the braid group as the fundamental group of the space of complex monic degree d polynomials with no multiple roots and with the presentation of the fundamental group of certain ordered configuration spaces over the reals which appear in the work of Khovanov. We further show that the groups pi(1)(P-d(c Theta)) admit an interpretation as special bordisms of immersions of one-manifolds into the cylinder R x S-1, whose images avoid the tangency patterns from Theta with respect to the generators of the cylinder.