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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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