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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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##### References:
  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  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  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  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  Cox, D. A., The functor of a smooth toric variety, Tohoku Math. J., 47, 251-262, (1995) · Zbl 0828.14035  Cox, D. A.; Little, J. B.; Schenck, H. K., Toric varieties, Grad. Stud. Math., vol. 124, (2011), Amer. Math. Soc. · Zbl 1223.14001  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  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  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  Guest, M. A., The topology of the space of rational curves on a toric variety, Acta Math., 174, 119-145, (1995) · Zbl 0826.14035  Guest, M. A.; Kozlowski, A.; Yamaguchi, K., The topology of spaces of coprime polynomials, Math. Z., 217, 435-446, (1994) · Zbl 0861.55015  Guest, M. A.; Kozlowski, A.; Yamaguchi, K., Spaces of polynomials with roots of bounded multiplicity, Fundam. Math., 116, 93-117, (1999) · Zbl 1016.55004  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  Kozlowski, A.; Yamaguchi, K., Simplicial resolutions and spaces of algebraic maps between real projective spaces, Topol. Appl., 160, 87-98, (2013) · Zbl 1276.55012  Mostovoy, J., Spaces of rational maps and the stone-Weierstrass theorem, Topology, 45, 281-293, (2006) · Zbl 1086.58005  Mostovoy, J., Truncated simplicial resolutions and spaces of rational maps, Q. J. Math., 63, 181-187, (2012) · Zbl 1237.58012  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  Segal, G. B., The topology of spaces of rational functions, Acta Math., 143, 39-72, (1979) · Zbl 0427.55006  Snaith, V. P., A stable decomposition of $$\operatorname{\Omega}^n S^n X$$, J. Lond. Math. Soc., 2, 577-583, (1974) · Zbl 0275.55019  Vassiliev, V. A., Complements of discriminants of smooth maps, (Topology and Applications, Transl. Math. Monogr., vol. 98, (1992), Amer. Math. Soc.), revised edition 1994  Vassiliev, V. A., Topologia dopolneniy k diskriminantam, (1997), Fazis Moskva
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