On manifolds with corners

TL;DR

This paper develops a comprehensive and well-behaved theory of manifolds with corners, introducing a new notion of smooth maps, with applications in Symplectic Geometry and moduli spaces of J-holomorphic curves.

Contribution

It presents a new, consistent framework for manifolds with corners, including a novel definition of smooth maps, ensuring better categorical properties and functorial boundaries.

Findings

The category of manifolds with corners has products and fiber products.

Boundaries behave functorially within the new framework.

The theory is tailored for applications in Symplectic Geometry.

Abstract

Manifolds without boundary, and manifolds with boundary, are universally known in Differential Geometry, but manifolds with corners (locally modelled on [0,\infty)^k x R^{n-k}) have received comparatively little attention. The basic definitions in the subject are not agreed upon, there are several inequivalent definitions in use of manifolds with corners, of boundary, and of smooth map, depending on the applications in mind. We present a theory of manifolds with corners which includes a new notion of smooth map f : X --> Y. Compared to other definitions, our theory has the advantage of giving a category Man^c of manifolds with corners which is particularly well behaved as a category: it has products and direct products, boundaries behave in a functorial way, and there are simple conditions for the existence of fibre products X x_Z Y in Man^c. Our theory is tailored to future applications in Symplectic Geometry, and is part of a project to describe the geometric structure on moduli spaces of J-holomorphic curves in a new way. But we have written it as a separate paper as we believe it is of independent interest.