Approximation and homotopy in regulous geometry

2023/11/09 by Wojciech Kucharz
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.1112/s0010437x23007522

Abstract

Let X , Y be nonsingular real algebraic sets. A map φ \colon X → Y is said to be k -regulous, where k is a nonnegative integer, if it is of class \mathcal C^k and the restriction of φ to some Zariski open dense subset of X is a regular map. Assuming that Y is uniformly rational, and k ≥ 1 , we prove that a \mathcal C^∞ map f \colon X → Y can be approximated by k -regulous maps in the \mathcal C^k topology if and only if f is homotopic to a k -regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking Y=\mathbb S^p (the unit p -dimensional sphere), we obtain several new results on approximation of \mathcal C^∞ maps from X into \mathbb S^p by k -regulous maps in the \mathcal C^k

Citations

• Approximation of maps into spheres by regulous maps

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