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Algebraic obstructions and a complete solution of a rational retraction problem


Author: Riccardo Ghiloni
Journal: Proc. Amer. Math. Soc. 130 (2002), 3525-3535
MSC (2000): Primary 14P05; Secondary 14P20, 14P25
DOI: https://doi.org/10.1090/S0002-9939-02-06617-0
Published electronically: May 15, 2002
MathSciNet review: 1918829
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Abstract: For each compact smooth manifold $W$ containing at least two points we prove the existence of a compact nonsingular algebraic set $Z$ and a smooth map $g: Z \longrightarrow W$ such that, for every rational diffeomorphism $r:Z'\longrightarrow Z$ and for every diffeomorphism $s: W' \longrightarrow W$ where $Z'$ and $W'$ are compact nonsingular algebraic sets, we may fix a neighborhood $\mathcal{U}$ of $s^{-1} \circ g \circ r$ in $C^{\infty}(Z',W')$ which does not contain any regular rational map. Furthermore $s^{-1} \circ g \circ r$ is not homotopic to any regular rational map. Bearing in mind the case in which $W$ is a compact nonsingular algebraic set with totally algebraic homology, the previous result establishes a clear distinction between the property of a smooth map $f$ to represent an algebraic unoriented bordism class and the property of $f$ to be homotopic to a regular rational map. Furthermore we have: every compact Nash submanifold of $\mathbb{R}^n$ containing at least two points has not any tubular neighborhood with rational retraction.


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Additional Information

Riccardo Ghiloni
Affiliation: Dipartimento di Matematica, University of Pisa, via Buonarroti 2, 56127 Pisa, Italy
Email: ghiloni@mail.dm.unipi.it

DOI: https://doi.org/10.1090/S0002-9939-02-06617-0
Keywords: Algebraic obstructions, regular rational retractions
Received by editor(s): August 1, 2001
Published electronically: May 15, 2002
Communicated by: Paul Goerss
Article copyright: © Copyright 2002 American Mathematical Society