[Paper Review] Localization Formulae in odd K-theory

Florentiu Daniel Cibotaru|arXiv (Cornell University)|Sep 15, 2022

Advanced Algebra and Geometry31 references3 citations

TL;DR

This paper introduces a geometric framework for odd K-theory using Grassmannians of Lagrangian subspaces in complex Hilbert spaces, where Schubert varieties serve as geometric representatives for cohomology ring generators. It establishes that the spectral flow corresponds to the first generator, providing a localization formula for the Poincaré duals of pull-back classes from families of self-adjoint Fredholm operators on closed manifolds.

ABSTRACT

We describe a class of real Banach manifolds, which classify $K^{-1}$. These manifolds are Grassmannians of (hermitian) lagrangian subspaces in a complex Hilbert space. Certain finite codimensional real subvarieties described by incidence relations define geometric representatives for the generators of the cohomology rings of these classifying spaces. Any family of self-adjoint, Fredholm operators parametrized by a closed manifold comes with a map to one of these spaces. We use these Schubert varieties to describe the Poincare duals of the pull-backs to the parameter space of the cohomology ring generators. The class corresponding to the first generator is the spectral flow.

Motivation & Objective

• To construct a geometric model for the classifying space of K^{-1} using Lagrangian Grassmannians in complex Hilbert spaces.
• To identify Schubert varieties of finite codimension as geometric representatives for cohomology ring generators of these classifying spaces.
• To provide a localization formula for the Poincaré duals of pull-back classes arising from families of self-adjoint Fredholm operators on closed manifolds.
• To characterize the first cohomology generator as the spectral flow invariant in this framework.

Proposed method

• Real Banach manifolds are constructed as Grassmannians of (hermitian) Lagrangian subspaces in a complex Hilbert space, serving as classifying spaces for K^{-1}.
• Finite codimensional real subvarieties defined by incidence relations—Schubert varieties—are used to represent cohomology ring generators.
• A family of self-adjoint Fredholm operators over a closed manifold induces a classifying map to one of these Grassmannian spaces.
• The Poincaré duals of the pull-backs of cohomology generators are realized geometrically via the homology classes of these Schubert varieties.
• The spectral flow invariant is identified as the class corresponding to the first generator of the cohomology ring.

Experimental results

Research questions

• RQ1 How can the classifying space for K^{-1} be realized geometrically using infinite-dimensional Grassmannians?
• RQ2 What role do Schubert varieties of finite codimension play in representing cohomology ring generators for these classifying spaces?
• RQ3 How can the Poincaré duals of pull-back classes from operator families be described using geometric cycles?
• RQ4 What is the geometric interpretation of the spectral flow in terms of K-theory and cohomology?

Key findings

• The Grassmannian of Lagrangian subspaces in a complex Hilbert space provides a classifying space for K^{-1}.
• Schubert varieties defined by incidence relations serve as geometric representatives for the cohomology ring generators of the classifying space.
• The spectral flow is identified as the class corresponding to the first generator of the cohomology ring.
• The Poincaré duals of pull-back classes from families of self-adjoint Fredholm operators are represented by the homology classes of these Schubert varieties.