 
Summary: ON THE FUNCTORIALITY OF THE BLOWUP
CONSTRUCTION.
GREGORY ARONE AND MARJA KANKAANRINTA
Abstract. We show a (slightly) new way to construct the blowup of a smooth
(or real analytic) manifold at a closed submanifold, and we use it to prove the
following functoriality property of the blowup: Let M and N be smooth (real
analytic) manifolds, with submanifolds A and B respectively. Let f : M N
be a smooth (real analytic) function such that f1
(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 lifts to a smooth (real analytic) map between the blowups
in a functorial way.
1. Introduction
The blowup of a smooth (or real analytic) manifold at a submanifold is an im
portant construction in geometry and topology. While the construction and its
basic properties are wellknown, they also seem to be somewhat folklore. Since
each one of the present authors had an occasion to be frustrated with the search
for a reference to a proof of the basic properties of the blowup construction (es
pecially properties having to do with functoriality in the smooth, or real analytic,
category), we decided to write such a reference ourselves.
