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december 2010 On the functoriality of the blow-up construction
Gregory Arone, Marja Kankaanrinta
Bull. Belg. Math. Soc. Simon Stevin 17(5): 821-832 (december 2010). DOI: 10.36045/bbms/1292334057

Abstract

We describe an explicit model for the blow-up construction in the smooth (or real analytic) category. We use it to prove the following functoriality property of the blow-up: Let $M$ and $N$ be smooth (real analytic) manifolds, with submanifolds $A$ and $B$ respectively. Let $f\colon M\to N$ be a smooth (real analytic) function such that $f^{-1}(B)=A$, and such that $f$ induces a fiberwise injective map from the normal space of $A$ to the normal space of $B$. Then $f$ has a unique lift to a smooth (real analytic) map between the blow-ups. In this way, the blow-up construction defines a continuous functor. As an application, we show how an action of a Lie group on a manifold lifts, under minimal hypotheses, to an action on a blow-up.

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Gregory Arone. Marja Kankaanrinta. "On the functoriality of the blow-up construction." Bull. Belg. Math. Soc. Simon Stevin 17 (5) 821 - 832, december 2010. https://doi.org/10.36045/bbms/1292334057

Information

Published: december 2010
First available in Project Euclid: 14 December 2010

zbMATH: 1252.57011
MathSciNet: MR2777772
Digital Object Identifier: 10.36045/bbms/1292334057

Subjects:
Primary: 57R35

Keywords: Blow-up , functorial , Lie group , proper action , real analytic , smooth

Rights: Copyright © 2010 The Belgian Mathematical Society

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Vol.17 • No. 5 • december 2010
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