Complements of resultants and homotopy types (Q1570024)

Let \(Q^d_{(n)}({\mathbb R})\) be the space consisting of \(n\)-tuples of monic degree \(d\) polynomials over the reals with no common real roots. For \(n\neq 3\), the homotopy type of \(Q^d_{(n)}({\mathbb R})\) is described in [\textit{A. Kozlowski} and \textit{K. Yamaguchi}, J. Math. Soc. Japan 52, 949-959 (2000)]. The present paper deals with the case \(n=3\). Let \(S^k\) denote the \(k\)-sphere and let \(\Omega X\) denote the loop space of \(X\); then there is a map from \(Q^d_{(3)}({\mathbb R})\) to \(\Omega S^2\) which is a homotopy equivalence up to dimension \(d\). If \(d\) is odd, say \(d=2m+1\), then \(Q^d_{(3)}({\mathbb R})\) is homotopy equivalent to \(S^1 \times J_m(\Omega S^3)\), where \(J_m(\Omega S^3)\) is the \(m\)th stage of the James filtration of \(\Omega S^3\). If \(d\) is even, say \(d=2m\), then after one suspension \(Q^d_{(3)}({\mathbb R})\) becomes homotopy equivalent to the \(2m\)-skeleton of \(\Omega S^2\). As a corollary, there is a computation of the cohomology ring of \(Q^d_{(3)}({\mathbb R})\) for odd \(d\).