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 \(D^\ast(d,n)=(2n-3)(d+1)-1\).
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 \(D^\ast(d,n)=(2n-3)(d+1)-1\).
Reviewer: Jelena Grbic (Manchester)
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 |
Keywords:
coordinate subspace; polyhedral product; fan; toric variety; primitive generator; holomorphic map; homotopy equivalence; simplicial resolution; Vassiliev spectral sequenceReferences:
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