Immersions and the unbounded Kasparov product: embedding spheres into Euclidean space

Journal of noncommutative geometry

article

Walter D. van Suijlekom¹  Luuk S. Verhoeven² 

[1] Radboud University Nijmegen;University of Western Ontario

关键词: KK-theory;    unboundedKK-theory;    Kasparov product;    immersions of manifolds;   

DOI  :  10.4171/jncg/451

学科分类：神经科学

来源: European Mathematical Society

【 摘 要 】

We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded KKKKKK-cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on Rn+1\mathbb{R}^{n+1}Rn+1. We find that the resulting spectral triple for the algebra C(Sn)C(\mathbb{S}^n)C(Sn) differs from the Dirac operator on the round sphere by a so-called index cycle, whose class in KK0(C,C)KK_0(\mathbb{C},\mathbb{C})KK0​(C,C) represents the multiplicative unit. At all points we check that our construction involving the unbounded Kasparov product is compatible with the bounded Kasparov product using Kucerovsky’s criterion and we thus capture the composition law for the shriek map for these immersions at the unbounded KKKKKK-theoretical level, while retaining the geometric information.

【 授权许可】

CC BY