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The topology of spaces of coprime polynomials. (English) Zbl 0861.55015
Let $$E^n_d$$ denote the set of $$n$$-tuples of mutually coprime complex polynomials of degree $$d$$. Translating a polynomial into its set of roots – considered as a positive divisor in $$\mathbb{C}$$ – one may, for example, envision these spaces as subsets of the $$n$$-fold product of the $$d$$-fold symmetric product $$\mathbb{C}$$, the complex numbers.
The main result is to construct a map $$E^n_d\to\Omega^2_0(\bigvee^n_{i=1}\mathbb{C} P^\infty)$$ which is a homotopy equivalence up to dimension $$d$$. Similar results are obtained for the space of holomorphic maps from the Riemann sphere into complements of unions of hyperplanes in $$\mathbb{C} P^n$$.

##### MSC:
 55Q52 Homotopy groups of special spaces 57T99 Homology and homotopy of topological groups and related structures
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