Algebraic obstructions and a complete solution of a rational retraction problem
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- by Riccardo Ghiloni
- Proc. Amer. Math. Soc. 130 (2002), 3525-3535
- DOI: https://doi.org/10.1090/S0002-9939-02-06617-0
- Published electronically: May 15, 2002
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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.References
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Bibliographic Information
- Riccardo Ghiloni
- Affiliation: Dipartimento di Matematica, University of Pisa, via Buonarroti 2, 56127 Pisa, Italy
- MR Author ID: 699436
- Email: ghiloni@mail.dm.unipi.it
- Received by editor(s): August 1, 2001
- Published electronically: May 15, 2002
- Communicated by: Paul Goerss
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1918829