Functorial compactification of linear spaces

TL;DR

The paper introduces a functorial approach to compactifying vector spaces into manifolds with corners, enabling the extension of linear maps to b-maps and b-fibrations, with applications to iterated blow-ups.

Contribution

It develops a new functorial compactification framework for linear spaces, extending Melrose's b-calculus and providing criteria for b-fibrations.

Findings

Compactifications are manifolds with corners.

Linear maps lift to b-maps and sometimes to b-fibrations.

The theory applies to iterated blow-ups of intersecting submanifolds.

Abstract

We define compactifications of vector spaces which are functorial with respect to certain linear maps. These "many-body" compactifications are manifolds with corners, and the linear maps lift to b-maps in the sense of Melrose. We derive a simple criterion under which the lifted maps are in fact b-fibrations, and identify how these restrict to boundary hypersurfaces. This theory is an application of a general result on the iterated blow-up of cleanly intersecting submanifolds which extends related results in the literature.