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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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