# zbMATH — the first resource for mathematics

Complements of resultants and homotopy types. (English) Zbl 0957.55006
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 [A. Kozlowski and 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 $$d$$.

##### MSC:
 55P15 Classification of homotopy type 55Q52 Homotopy groups of special spaces
##### Keywords:
complement of resultant; homotopy type
Full Text: