The topology of spaces of coprime polynomials. (English) Zbl 0861.55015

Guest, M. A.; Kozlowski, A.; Yamaguchi, K.

Math. Z. 217, No. 3, 435-446 (1994).

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 \(\mathbb{C} P^n\).

Reviewer: V.P.Snaith (Hamilton/Ontario)

MSC:

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

Keywords:

complex polynomials; homotopy equivalence; space of holomorphic maps

References:

[1]	[Bj] Björner, A.: Subspace arrangementsinfo to appear in Proc. 1st European Congress of Mathematicians, Paris 1992 · Zbl 0778.05089
[2]	[Ep] Epshtein, S.I.: Fundamental groups of spaces of coprime polynomials. Functional Analysis and its Applications 7: 82–83 (1973) · Zbl 0312.55001 · doi:10.1007/BF01075662
[3]	[Fu] Fulton, W.: Introduction to Toric Varieties. Ann. of Math. Stud. 131. Princeton Univ. Press 1993 · Zbl 0813.14039
[4]	[GKY] Guest, M.A., Kozlowski, A., and Yamaguchi, K.: A new configuration space model for 3 S 3. Preprint
[5]	[Gu1] Guest, M. A.: Instantons, rational maps, and harmonic maps. Matemática Contemporânea 2: 113–155 (1992) · Zbl 0875.30005
[6]	[Gu2] Guest, M.A.: The topology of the space of rational curves on a toric variety. Preprint
[7]	[Ha] Hansen, V.L.: Braids and Coverings. L.M.S. Student Texts 18 Cambridge Univ. Press 1989
[8]	[HH] Hausmann, J.-C. and Husemoller, D.: Acyclic maps L’Enseignement Math. 25: 53–75 (1979) · Zbl 0412.55008
[9]	[Ko] Kozlowski, A.: Stabilization of homology groups of spaces of mutually disjoint divisors. RIMS (Kyoto University) Kokyuroku 8: 108–116 (1992)
[10]	[Mc] McDuff, D.: Configuration spaces of positive and negative particles. Topology 14: 91–107 (1975) · Zbl 0296.57001 · doi:10.1016/0040-9383(75)90038-5
[11]	[Od] Oda, T.: Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties. Berlin: Springer 1988 · Zbl 0628.52002
[12]	[Se] Segal, G.B.: The topology of spaces of rational functions. Acta Math. 143: 39–72 (1979) · Zbl 0427.55006 · doi:10.1007/BF02392088