Speaker: Jody Trout, Dartmouth College

Date: May 5, 2026

Abstract: We show that some well-known filtrations of infinite-dimensional groups of Fredholm operators associated to certain perturbation classes are, in fact, Fredholm $\Delta$-filtrations. For example, let $E$ be a separable infinite dimensional real Hilbert space. The group $GL_K(E)$ of all invertible operators on $E$ which are compact perturbations of the identity is the structure group for Hilbert Fredholm manifolds and bundles modelled on $E$. Using an orthonormal basis, there are canonical inclusions of general linear groups: \(GL(1) \subset \cdots \subset GL(n) \subset GL(n+1) \subset \cdots \subset GL(\infty) = \lim GL(n) \subset GL_K(E).\) We show this is a Fredholm $\Delta$-filtration of the Banach manifold $GL_K(E)$ with dimension sequence $\Delta(n) = \dim(\GL(n)) = n^2$, which was not discussed in the classical Fredholm manifold literature because of rigid dimension constraints. This is joint work with Ahmad Reza.