On Deformation Spaces, Tangent Groupoids and Generalized Filtrations of Banach and Fredholm Manifolds

TL;DR

This paper extends deformation and tangent groupoid constructions to infinite-dimensional Banach and Fredholm manifolds, introducing generalized filtrations that enable new functorial tangent bundle and groupoid structures.

Contribution

It generalizes deformation to the normal cone and tangent groupoid constructions to infinite-dimensional manifolds and introduces a flexible theory of generalized filtrations.

Findings

Constructed tangent groupoids for Banach and Fredholm manifolds.

Developed generalized filtrations compatible with tangent structures.

Enabled new functorial relationships in infinite-dimensional geometry.

Abstract

We extend the deformation to the normal cone and tangent groupoid constructions from finite-dimensional manifolds to infinite-dimensional Banach and Fredholm manifolds. Next, we generalize the concept of Fredholm filtrations to get a more flexible and functorial theory. In particular, we show that if $M$ is a Banach (or Fredholm) manifold with generalized filtration ${\mathcal F} = \{M_n\}_1^\infty$ by finite-dimensional submanifolds, then there are induced generalized filtrations $T{\mathcal F} = \{TM_n\}_1^\infty$ of the tangent bundle $TM$ and $\mathbb{T}{\mathcal F} = \{\mathbb{T}{M_n}\}_1^\infty$ of the tangent groupoid $\mathbb{T}{M}$, which is not possible in the classical theory.