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The homotopy type of spaces of coprime polynomials revisited. (English) Zbl 1387.55012
Let \(n\geq 2\) and \([n]=\{0,1,\dots,n-1\}\). For each subset \(\sigma=\{i_1,\dots,i_s\}\subset [n]\), let \(L_\sigma\subset\mathbb C^n\) denote the coordinate subspace in \(\mathbb C^n\) defined by \(L_\sigma=\{(x_0,x_1,\dots,x_{n-1})\in\mathbb C^n\mid x_{i_1}=\cdots=x_{i_s}=0\}\).
Let \(I\) be any collection of subsets of \([n]\) such that \(|\sigma|\geq 2\) for all \(\sigma\in I\). Let \(Y_I\subset\mathbb C^n\) be the complement of the arrangement of coordinate subspaces defined by \(Y_1=\mathbb C^n\backslash L(I)\), where \(L(I)=\cup_{\sigma\in I}L_\sigma\). Consider the natural free \(\mathbb C^\ast\)-action on \(Y_I\) given by coordinate-wise multiplication and let \(X_I\) denote the orbit space.
For \(I\) any collection of subsets of \([n]\) and \((X, \ast)\) a based space, let \(\vee ^IX\subset X^n\) denote the subspace consisting of all \((x_0,\dots,x_{n-1})\in X^n\) such that, for each \(\sigma\in I\), \(x_j=\ast\) for some \(j\in\sigma\). The space \(\vee^IX\) is called the generalized wedge product of \(X\) of type \(I\) and there is a homotopy equivalence \(\Omega^2_dX_I\simeq\Omega^2(\vee^I\mathbb CP^\infty)\).
The purpose of this paper is to study the topology of certain toric varieties \(X_I\) and to improve the classical homotopy stability dimension for the inclusion map \(i_d:\mathrm{Hol}^\ast_d (S^2,X_I)\to\mathrm{Map}^\ast_d(S^2,X_I)\) by making use of the Vassiliev spectral sequence. The authors also improve the homotopy stability dimension of this inclusion given by G. Segal for \(X_I=\mathbb CP^{n-1}\) and \(n\geq 3\).
Let \(r_{\min}(I)\) denote the positive integer defined by \(r_{\min}(I)=\min\{|\sigma|:\sigma\in I\}\).
The main results are:
a) If \(r_{\min} (I)\geq 3\), the inclusion map \[ i_d:\mathrm{Hol}^ \ast_d(S^2,X_I)\to\mathrm{Map}^\ast_d(S^2,X_I)=\Omega 2_dX_I\simeq\Omega^2(\vee^I\mathbb CP^\infty) \] is a homotopy equivalence through dimension \(D(I;d)=(2r_{ \min}(I)-3)d-2\).
b) (The case \(I=I(n)\)). If \(n\geq 3\), the inclusion map \[ i_d:\mathrm{Hol}^ \ast_d(S^2,\mathbb CP^{n-1})\to\mathrm{Map}^\ast_d(S^2,\mathbb CP^{n-1})=\Omega 2_d\mathbb CP^{n-1}\simeq\Omega^2S^{2n-1} \] is a homotopy equivalence through dimension \(D^\ast(d,n)=(2n-3)(d+1)-1\).

MSC:
55P10 Homotopy equivalences in algebraic topology
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55P35 Loop spaces
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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[1] Adamaszek, M.; Kozlowski, A.; Yamaguchi, K., Spaces of algebraic and continuous maps between real algebraic varieties, Q. J. Math., 62, 771-790, (2011) · Zbl 1245.14060
[2] Boyer, C. P.; Mann, B. M., Monopole, non-linear σ-models and two-fold loop spaces, Commun. Math. Phys., 115, 571-594, (1988) · Zbl 0656.58049
[3] Buchstaber, V. M.; Panov, T. E., Torus actions and their applications in topology and combinatorics, Univ. Lect. Note Ser., vol. 24, (2002), Amer. Math. Soc. Providence · Zbl 1012.52021
[4] Cohen, F. R.; Cohen, R. L.; Mann, B. M.; Milgram, R. J., The topology of rational functions and divisors of surfaces, Acta Math., 166, 163-221, (1991) · Zbl 0741.55005
[5] Cox, D. A., The functor of a smooth toric variety, Tohoku Math. J., 47, 251-262, (1995) · Zbl 0828.14035
[6] Cox, D. A.; Little, J. B.; Schenck, H. K., Toric varieties, Grad. Stud. Math., vol. 124, (2011), Amer. Math. Soc. · Zbl 1223.14001
[7] Cohen, F. R.; Mahowald, M. E.; Milgram, R. J., The stable decomposition for the double loop space of a sphere, Proc. Symp. Pure Math., 33, 225-228, (1978) · Zbl 0406.55007
[8] Cohen, F. R.; Moore, J. C.; Neisendorfer, J. A., The double suspension and exponents of the homotopy groups of spheres, Ann. Math., 110, 549-565, (1979) · Zbl 0443.55009
[9] Guest, M. A., On the space of holomorphic maps from the Riemann sphere to the quadric cone, Q. J. Math. Oxf., 45, 57-75, (1994) · Zbl 0802.58012
[10] Guest, M. A., The topology of the space of rational curves on a toric variety, Acta Math., 174, 119-145, (1995) · Zbl 0826.14035
[11] Guest, M. A.; Kozlowski, A.; Yamaguchi, K., The topology of spaces of coprime polynomials, Math. Z., 217, 435-446, (1994) · Zbl 0861.55015
[12] Guest, M. A.; Kozlowski, A.; Yamaguchi, K., Spaces of polynomials with roots of bounded multiplicity, Fundam. Math., 116, 93-117, (1999) · Zbl 1016.55004
[13] Kozlowski, A.; Ohno, M.; Yamaguchi, K., Spaces of algebraic maps from real projective spaces to toric varieties, J. Math. Soc. Jpn., 68, 2, 745-771, (2016) · Zbl 1353.55009
[14] Kozlowski, A.; Yamaguchi, K., Simplicial resolutions and spaces of algebraic maps between real projective spaces, Topol. Appl., 160, 87-98, (2013) · Zbl 1276.55012
[15] Mostovoy, J., Spaces of rational maps and the stone-Weierstrass theorem, Topology, 45, 281-293, (2006) · Zbl 1086.58005
[16] Mostovoy, J., Truncated simplicial resolutions and spaces of rational maps, Q. J. Math., 63, 181-187, (2012) · Zbl 1237.58012
[17] Mostovoy, J.; Munguia-Villanueva, E., Spaces of morphisms from a projective space to a toric variety, Rev. Colomb. Mat., 48, 41-53, (2014) · Zbl 1350.14037
[18] Segal, G. B., The topology of spaces of rational functions, Acta Math., 143, 39-72, (1979) · Zbl 0427.55006
[19] Snaith, V. P., A stable decomposition of \(\operatorname{\Omega}^n S^n X\), J. Lond. Math. Soc., 2, 577-583, (1974) · Zbl 0275.55019
[20] Vassiliev, V. A., Complements of discriminants of smooth maps, (Topology and Applications, Transl. Math. Monogr., vol. 98, (1992), Amer. Math. Soc.), revised edition 1994
[21] Vassiliev, V. A., Topologia dopolneniy k diskriminantam, (1997), Fazis Moskva
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