The homotopy type of the space of algebraic loops on a toric variety. (English) Zbl 1514.55007
For two topological spaces \(X\) and \(Y\) with base points, let Map\(^{*}(X,Y)\) denote the space of all continuous based maps \(f:X\to Y\) with the compact-open topology. When the spaces \(X\) and \(Y\) have additional structure, e.g. that of a complex or symplectic manifold or an algebraic variety, a natural question is to consider the subspace \(\mathcal{S}(X,Y)\) of Map\(^{*}(X,Y)\) of all based maps \(f\) which preserve this structure and to ask whether the inclusion map of \(\mathcal{S}(X,Y)\) into Map\(^{*}(X,Y)\) is a homotopy or homology equivalence up to some dimension. The authors make a contribution to this problem by studying the homotopy type of the space of tuples of polynomials inducing base-point preserving algebraic maps from the circle \(S^1\) to a toric variety \(X_{\Sigma}\) and proving amongst others a homotopy stability result for this space.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)
MSC:
| 55P15 | Classification of homotopy type |
| 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 |
| 57S12 | Toric topology |
Keywords:
polyhedral product; fan; toric variety; moment-angle complex; primitive generator; algebraic map; Vassiliev spectral sequenceReferences:
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