On Deformation Spaces, Tangent Groupoids and Generalized Filtrations of Banach and Fredholm Manifolds
Ahmad Reza Haj Saeedi Sadegh, Jody Trout
TL;DR
The paper extends key finite-dimensional constructions—deformation to the normal cone ${\operatorname{DNC}}$ and Connes' tangent groupoid ${\mathbb{T}}{M}$—to Banach and Fredholm manifolds, proving these spaces retain smooth, functorial structure and compatible transversality. It introduces canonical Fredholm structures on ${\operatorname{DNC}}(M,M_0)$ and on ${\mathbb{T}}{M}$, and proves that maps of index zero preserve Fredholmness under deformation. The central advance is a theory of generalized $\Delta$-filtrations for Banach and Fredholm manifolds, including their functorial behavior under products and tangent constructions, yielding induced filtrations ${T}{\mathcal{F}}$ and ${\mathbb{T}}{\mathcal{F}}$ that overcome limitations of classical filtrations. These developments lay groundwork for infinite-dimensional index theory, Lie groupoid methods, and noncommutative geometric approaches by enabling structured filtrations on tangent structures and deformations.
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.