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ON THE FUNCTORIALITY OF THE BLOW-UP CONSTRUCTION.
 

Summary: ON THE FUNCTORIALITY OF THE BLOW-UP
CONSTRUCTION.
GREGORY ARONE AND MARJA KANKAANRINTA
Abstract. We show a (slightly) new way to construct the blow-up of a smooth
(or real analytic) manifold at a closed submanifold, and 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 : M 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 lifts to a smooth (real analytic) map between the blow-ups
in a functorial way.
1. Introduction
The blow-up 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 well-known, 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 blow-up construction (es-
pecially properties having to do with functoriality in the smooth, or real analytic,
category), we decided to write such a reference ourselves.

  

Source: Arone, Gregory - Department of Mathematics, University of Virginia

 

Collections: Mathematics