1 results
Approximation and homotopy in regulous geometry
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 09 November 2023, pp. 1-20
- Print publication:
- January 2024
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- Article
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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 
$k \geq 0$.
