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Approximation and homotopy in regulous geometry

Part of: Polynomials, rational functions Differential topology

Published online by Cambridge University Press:  09 November 2023

Wojciech Kucharz Open the ORCID record for Wojciech Kucharz [Opens in a new window]
Wojciech Kucharz*
Affiliation:
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland Wojciech.Kucharz@im.uj.edu.pl
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Abstract

Let $X$, $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$-regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$, we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ 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 $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$-regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$.

Keywords

real algebraic varietyregular mapk-regulous mapapproximationhomotopymalleable varietyunit sphere

MSC classification

Primary: 14P05: Real algebraic sets 14P25: Topology of real algebraic varieties 26C15: Rational functions 57R99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 1 , January 2024 , pp. 1 - 20
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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