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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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