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Stable splitting of the space of polynomials with roots of bounded multiplicity. (English) Zbl 0917.55005
Let \(D_j=F({\mathbb C},j)_+\wedge_{\Sigma_j} S^j\) be the \(j\)th subquotient of the May-Milgram model for \(\Omega^2S^3\), where \(F({\mathbb C},j)_+\) denotes the configuration space \(F({\mathbb C},j)\) of \(j\) distinct (ordered) points in the complex plane \({\mathbb C}\) together with an added disjoint base point \(\{*\}\), and \(\Sigma_j\) is the symmetric group on \(j\) letters which acts on both \(F({\mathbb C},j)\) and the \(j\)-sphere \(S^j=S^1\wedge S^1\dots \wedge S^1\). Let \(SP^d_n({\mathbb C})\) be the space of all monic complex polynomials of degree \(d\) without multiplicity \(\geq n\). Finally, let Hol\(^*_d(S^2,{\mathbb C}P^{n-1})\) denote the space of basepoint preserving holomorphic maps of degree \(d\) from the Riemann sphere \(S^2={\mathbb C}\cup\infty\) to the complex projective space \({\mathbb C}P^{n-1}\). Then F. R. Cohen, R. L. Cohen, B. M. Mann, and R. J. Milgram [Acta Math. 166, No. 3/4, 163-221 (1991; Zbl 0741.55005)] proved that there is a stable homotopy equivalence Hol\(^*_d(S^2,{\mathbb C}P^{n-1})\simeq_s \bigwedge_{j=1}^d\Sigma^{2(n-2)j}D_j\), where \(\Sigma^k\) denotes the \(k\) fold reduced suspension, and V. A. Vassiliev [Complements of discriminants of smooth maps: topology and applications, Transl. Math. Monogr. 98 (1992; Zbl 0762.55001)] proved that there is a stable homotopy equivalence \(\text{Hol}^*_d(S^2,{\mathbb C}P^{n-1})\simeq_s SP^{dn}_n({\mathbb C})\). By combining these results, one obtains a stable homotopy equivalence \(\bigwedge_{j=1}^d\Sigma^{2(n-2)j}D_j\simeq_s SP^{dn}_n({\mathbb C})\). In the present paper, the authors provide a direct proof of this interesting stable homotopy equivalence, basically by imitating the method used by F. R. Cohen, R. L. Cohen, B. M. Mann, and R. J. Milgram [Math. Z. 213, No. 1, 31-47 (1993; Zbl 0790.55005)] replacing \(\text{Hol}^*_d(S^2,{\mathbb C}P^{n-1})\) by \(SP^{dn}_n({\mathbb C})\). As applications, the present authors prove, among others, a stronger version of Vassiliev’s theorem.

MSC:
55P42 Stable homotopy theory, spectra
55P15 Classification of homotopy type
55P35 Loop spaces
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