Bulletin of the Belgian Mathematical Society - Simon Stevin

On the functoriality of the blow-up construction

Gregory Arone and Marja Kankaanrinta

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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.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 5 (2010), 821-832.

First available in Project Euclid: 14 December 2010

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Zentralblatt MATH identifier

Primary: 57R35: Differentiable mappings

blow-up functorial real analytic smooth Lie group proper action


Arone, Gregory; Kankaanrinta, Marja. On the functoriality of the blow-up construction. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 5, 821--832. doi:10.36045/bbms/1292334057. https://projecteuclid.org/euclid.bbms/1292334057

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