Problem 3 — Is the complement of the essential ramification set in ${\mathbb{R}}^{d}$ a $K(\pi,1)$-space?

Definitions and mathematical statement

Let $E_d\subseteq{\mathbb{R}}^d$ be the essential ramification set for the real-polynomial family intended in the source, and put $X_d={\mathbb{R}}^d\setminus E_d$. Vassiliev asks whether $X_d$ is a $K(\pi,1)$-space. The defining polynomial family, coefficient coordinates, permitted $d$, and definition of essential ramification are absent from this excerpt; $E_d$ is a reference to that specific missing set, not the ordinary discriminant by default.

A path-connected space of CW type is a $K(\pi,1)$ when its fundamental group is $\pi$ and \[\pi_j(X_d,x)=0\qquad(j\geq2),\] with the usual local hypotheses giving the equivalent condition that its universal cover is contractible. If $X_d$ is disconnected, the assertion must either specify a component or mean that every component is aspherical; a single connected Eilenberg--Mac Lane space and a disconnected union of such components are different conventions. The excerpt does not resolve this point. Thus a precise formulation, once $E_d$ and the component convention are fixed, asks for vanishing of all higher homotopy groups, not merely a computation of $\pi_1$ or homology.

The supplied context distinguishes this essential-ramification complement from classical discriminant/bifurcation complements, including the simple-singularity discriminant examples. Results about those other complements do not identify or settle this space without a comparison.

Short English statement

Is the complement of the specified essential ramification locus aspherical? The locus, degree range and connected-component convention are not defined in the short record and cannot be replaced by an ordinary polynomial discriminant.

Sources

• Local UnsolvedMath registry, permanent ID $10800009$, exact source code catalogue entry.
• Victor Vassiliev, A Few Problems on Monodromy and Discriminants, Problem 3: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/. Registry access 2026-07-29; html-structure extraction.
• V. A. Vassiliev, On topological invariants of real algebraic functions, Functional Analysis and Its Applications 45:3 (2011), 163--172; identified by the summary as source reference [essent].
• The summary mentions Looijenga (1974) for simple-singularity discriminant complements, without a full entry, as context about a different space.