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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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