[Paper Review] Generalized Riemann-Hilbert Transmission and Boundary Value Problems, Fredholm Pairs and Bordisms

Bogdan Bojarski, Andrzej Weber|ArXiv.org|Nov 7, 2001

Advanced Operator Algebra Research11 references3 citations

TL;DR

This paper introduces an abstract framework for generalized Riemann-Hilbert transmission problems using Fredholm pairs in Hilbert spaces with splittings, establishing a connection between boundary value problems, K-theory, and bordism theory. The key contribution is a local-to-global index formula for elliptic operators on manifolds via Fredholm bordisms, where the global index equals the sum of partial indices associated with boundary data and splittings.

ABSTRACT

We present classical and generalized Riemann-Hilbert problem in several contexts arising from $K$-theory and bordism theory. The language of Fredholm pairs turns out to be useful and unavoidable. We propose an abstract formulation of a notion of bordism in the context of Hilbert spaces equipped with splittings.

Motivation & Objective

• To develop an abstract functional-analytic framework for Riemann-Hilbert transmission problems using Fredholm pairs in Hilbert spaces.
• To establish a correspondence between geometric bordisms and Fredholm bordisms in Hilbert spaces equipped with splittings.
• To generalize the Atiyah-Singer index theorem to boundary value problems via a local-to-global index decomposition.
• To connect the theory of elliptic operators on manifolds with K-theory and bordism through the index of Fredholm pairs.
• To formalize the role of Calderón projectors and boundary value spaces in defining Fredholm bordisms and computing indices.

Proposed method

• Uses the language of Fredholm pairs (H⁻, H⁺) in Hilbert spaces, where H⁻ + H⁺ is closed and dim(H⁻ ∩ H⁺), codim(H⁻ + H⁺) are finite.
• Introduces a bordism category on Hilbert spaces with splittings (H⁷, H⁷), where Fredholm pairs represent bordisms between boundary data.
• Defines the index of a subspace L ⊂ H₁ ⊕ H₂ with respect to splittings via the formula κ(H⁷₁|L|H⁷₂) = dim(L ∩ (H⁷₁ ⊕ H⁷₂)) − codim(L + (H⁷₁ ⊕ H⁷₂)).
• Applies the theory to Riemann surfaces with boundary, modeling boundary values of holomorphic functions on punctured surfaces.
• Computes the index of the Cauchy-Riemann operator on a genus-g surface with k incoming and l outgoing boundary components as 1 − g − ∑λᵢ + ∑μⱼ − l.
• Establishes a composition law for bordisms and shows that the global index equals the sum of partial indices along a decomposition of a manifold into bordisms.

Experimental results

Research questions

• RQ1 How can generalized Riemann-Hilbert problems be formulated abstractly in terms of Fredholm pairs in Hilbert spaces?
• RQ2 What is the relationship between geometric bordism and Fredholm bordism in the context of Hilbert spaces with splittings?
• RQ3 How does the index of a boundary value problem relate to topological invariants of the underlying manifold?
• RQ4 Can a local-to-global index formula be derived for elliptic operators on manifolds with boundary using Fredholm pairs?
• RQ5 What is the role of Calderón projectors and boundary value spaces in defining Fredholm bordisms and computing indices?

Key findings

• The index of the Cauchy-Riemann operator on a genus-g Riemann surface with k holomorphic disks and l anti-holomorphic disks is 1 − g, computed via the Fredholm index of boundary value spaces.
• For a general elliptic operator D on a manifold decomposed into bordisms, the global index equals the sum of partial indices: ind D = ∑ᵢ κ(H⁷ᵢ|Lᵢ|H⁷ᵢ₊₁).
• The index κ(H⁷₁|L|H⁷₂) is invariant under small perturbations of the pair (H⁻, H⁺), generalizing Kato’s stability result for Fredholm pairs.
• A Fredholm bordism arises from the graph of an isomorphism in GL(S, K), and its index equals the K-theoretic invariant κ(ϕ) of the isomorphism.
• When composing bordisms with matching splittings, the total index may differ from the sum of partial indices by a defect equal to the index on any closed component formed.
• The theory provides a functional-analytic counterpart to geometric bordism, where Hilbert spaces with splittings and Fredholm pairs form a category isomorphic to the bordism category.