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Riemannian embeddings in codimension one as unbounded $KK$-cycles

Walter D. van Suijlekom and Luuk S. Verhoeven

Vol. 8 (2023), No. 4, 645–668
Abstract

Given a codimension one Riemannian embedding of Riemannian spinc-manifolds ı : X Y we construct a family {ı!𝜖}0<𝜖<𝜖0 of unbounded KK-cycles from C(X) to C0(Y ), each equipped with a connection 𝜖 and each representing the shriek class ı! KK(C(X),C0(Y )). We compute the unbounded product of ı!𝜖 with the Dirac operator DY on Y and show that this represents the KK-theoretic factorization of the fundamental class [X] = ı! [Y ] for all 𝜖. In the limit 𝜖 0 the product operator admits an asymptotic expansion of the form 1 𝜖T + DX + (𝜖) where the “divergent” part T is an index cycle representing the unit in KK(, ) and the constant “renormalized” term is the Dirac operator DX on X. The curvature of (ı!𝜖,𝜖) is further shown to converge to the square of the mean curvature of ı as 𝜖 0.

Keywords
noncommutative geometry, spectral triple, unbounded product, unbounded KK-theory, Riemannian immersion
Mathematical Subject Classification
Primary: 58B34
Secondary: 53C21
Milestones
Received: 19 April 2023
Revised: 11 July 2023
Accepted: 3 September 2023
Published: 6 December 2023
Authors
Walter D. van Suijlekom
IMAPP
Radboud University
Heyendaalseweg 135
6525 AJ Nijmegen
Netherlands
Luuk S. Verhoeven
University of Western Ontario
Middlesex College
London, ON
Canada