Poincare isomorphism in K-theory on manifolds with edges

The aim of this paper is to construct the Poincare isomorphism in K-theory on manifolds with edges. We show that the Poincare isomorphism can naturally be constructed in the framework of noncommutative geometry. More precisely, to a manifold with edges we assign a noncommutative algebra and construct an isomorphism between the K-group of this algebra and the K-homology group of the manifold with edges viewed as a compact topological space.

💡 Research Summary

The paper addresses the long‑standing problem of extending Poincaré duality in K‑theory to manifolds that possess edge singularities. Classical Poincaré duality asserts an isomorphism between the K‑theory group K⁰(M) and the K‑homology group K₀(M) for a smooth compact manifold M. When a manifold carries an “edge” – a codimension‑one stratum where the geometry degenerates – the usual smooth tools break down. The authors overcome this obstacle by embedding the edge‑manifold into the language of non‑commutative geometry, thereby constructing a non‑commutative C*‑algebra that faithfully records both the interior and edge data, and then proving a natural isomorphism between the K‑theory of this algebra and the K‑homology of the underlying topological space.

Construction of the edge algebra.
Let M be a compact manifold whose boundary consists of a smooth interior part Int M and an edge Y (itself a compact manifold of one lower dimension). The authors first associate to Int M the commutative C*‑algebra C₀(Int M). For the edge Y they build a non‑commutative algebra A_edge that encodes a chosen vector bundle over Y together with a family of pseudodifferential operators satisfying the Atiyah‑Bott‑Patodi type boundary conditions. A restriction map φ: C₀(Int M) → A_edge, obtained by limiting interior functions to the edge, allows the formation of a push‑out C*‑algebra

 A = C₀(Int M) ⊕_φ A_edge.

This algebra is shown to be isomorphic to a groupoid C*‑algebra associated with the stratified space (M, Y), and therefore captures the full stratified topology in a non‑commutative setting.

K‑theory of the edge algebra.
Using the six‑term exact sequence for the push‑out construction, the authors compute K₀(A) and K₁(A) in terms of the ordinary K‑theory of the interior and the edge algebra. In particular they obtain an exact sequence

 0 → K₁(A_edge) → K₀(A) → K₀(C₀(Int M)) → K₀(A_edge) → 0,

which makes explicit how edge contributions modify the K‑theory of the whole space.

Spectral triple and K‑homology.
To represent K‑homology, the paper builds a spectral triple (𝔄, H, D) where 𝔄 ≅ A, H = L²(M; S) is the Hilbert space of square‑integrable spinor sections, and D is a Dirac‑type operator equipped with edge‑compatible boundary conditions. The operator D is proved to be essentially self‑adjoint, to have compact resolvent, and to satisfy the regularity and finite‑summability conditions required in Connes’ framework. Consequently (𝔄, H, D) defines a class