On the functoriality of the blow-up construction

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.