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Topology of complements of discriminants and resultants. (English) Zbl 0974.55002
Let $$P^d_n({\mathbb C})$$ denote the space of all monic polynomials over $${\mathbb C}$$ of degree $$d$$ that have no repeated real roots (complex roots may be repeated). This type of space, a so-called complement of a discriminant, is of current interest in several fields of mathematics [V. A. Vassiliev, Proc. Internat. Congr. Math. ICM ’94 Zürich, 209-226 (1995; Zbl 0852.55003)]. It is shown here that $$P^d_n({\mathbb C})$$ is homotopy equivalent to $$J_{[d/n]}(\Omega S^{2n-1})$$ for $$n \geq 2$$, where $$[d/n]$$ denotes the integer part of $$d/n$$ and $$J_{m}(\Omega S^{2n-1})$$ denotes the $$m$$th James filtration of the loop space $$\Omega S^{2n-1}$$. It follows that $$P^d_n({\mathbb C})$$ can be regarded as a finite-dimensional approximation to the (infinite-dimensional) loop space $$\Omega S^{2n-1}$$. Several other interesting results of a similar nature are proved.
The work presented here relates to various other results. For instance, let $$P^d_n({\mathbb R})$$ denote the space of all monic polynomials over $${\mathbb R}$$ of degree $$d$$ that have no repeated real roots. Vassiliev has shown that $$P^d_n({\mathbb R})$$ and $$J_{m}(\Omega S^{n-1})$$ are homotopy equivalent for $$n \geq 4$$ (quoted as Theorem 1.2 of the current paper). The authors deduce this from their result previously mentioned, by restricting to the fixed point sets of a natural $${\mathbb Z}_2$$ action on the spaces concerned.

##### MSC:
 55P15 Classification of homotopy type 55P35 Loop spaces 55P10 Homotopy equivalences in algebraic topology
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