Functoriality for symplectic and contact cutting, and equivariant radial-squared blowups

TL;DR

This paper formalizes Lerman's symplectic and contact cutting procedures as functors between categories, enabling systematic transformations of manifolds, forms, and submanifolds with applications to radial-squared blowups.

Contribution

It introduces a functorial framework for symplectic and contact cutting, extending the procedure to forms, distributions, and submanifolds, and constructs an inverse functor for radial-squared blowups.

Findings

Lerman's cutting is a functor between specific manifold categories.

The procedure applies to non-symplectic forms and non-contact distributions.

An inverse functor for radial-squared blowup is established.

Abstract

We exhibit Lerman's cutting procedure as a functor from the category of manifolds-with-boundary equipped with free circle actions near the boundary, with so-called equivariant transverse maps, to the category of manifolds and smooth maps. We then apply the cutting procedure to differential forms that are not necessarily symplectic, to distributions that are not necessarily contact, and to submanifolds. We obtain an inverse functor from so-called equivariant radial-squared blowup.