Immersions and the unbounded Kasparov product: embedding spheres into Euclidean space. (English) Zbl 1502.19004
For a \(K\)-oriented map \(f \colon X \to Y\) between spin\(^{c}\) manifolds \(X,Y\), there is a \(KK\)-element \(f! \in KK^{\dim X + \dim Y}(C(X), C(Y))\) so-called shriek map. A one of interesting property of the shriek map is a formula \([D_{X}] = f! \; \hat{\otimes}_{C(Y)} [D_{Y}]\), where \([D_{X}]\) (resp. \([D_{Y}]\)) is a fundamental class of \(X\) (resp. \(Y\)) which is represented by the Dirac operator \(D_{X}\) on \(X\) (resp. \(D_{Y}\) on \(Y\)). The shriek map \(f!\) is constructed by an unbounded character in [A. Connes and G. Skandalis, Publ. Res. Inst. Math. Sci. 20, 1139–1183 (1984; Zbl 0575.58030)], so it is natural to investigate how this formula can be realized concretely in terms of unbounded \(KK\)-cycles in the sense of [S. Baaj and P. Julg, C. R. Acad. Sci., Paris, Sér. I 296, 875–878 (1983; Zbl 0551.46041)]. For example, when \(f\) is a submersion and \(X\) and \(Y\) are compact, the factorization has been investigated in [J. Kaad and W. D. van Suijlekom, J. Noncommut. Geom. 12, No. 3, 1133–1159 (2018; Zbl 1405.19002)].
In this paper under review, the authors investigate the factorization problem for the embedding \(i \colon S^{n} \to \mathbb{R}^{n+1}\). Namely, for an unbounded operator \(\widetilde{S}\) represents \(i!\), the tensor sum \(D_{\times} = \widetilde{S} \otimes 1 + \gamma^{3} \otimes_{\nabla} D_{\mathbb{R}^{n+1}}\) represents both \(i! \otimes [D_{\mathbb{R}^{n+1}}]\) and \([D_{S^{n}}]\) (see Theorem 1).
In this paper under review, the authors investigate the factorization problem for the embedding \(i \colon S^{n} \to \mathbb{R}^{n+1}\). Namely, for an unbounded operator \(\widetilde{S}\) represents \(i!\), the tensor sum \(D_{\times} = \widetilde{S} \otimes 1 + \gamma^{3} \otimes_{\nabla} D_{\mathbb{R}^{n+1}}\) represents both \(i! \otimes [D_{\mathbb{R}^{n+1}}]\) and \([D_{S^{n}}]\) (see Theorem 1).
Reviewer: Tatsuki Seto (Tokyo)
References:
| [1] | Lemma A.1. Let .E; |
| [2] | D/ be an odd unbounded A-B KK-cycle. Then |
| [3] | E˝C 2 ; D˝ 2 I 1˝ 3 / is an even unbounded A˝Cl 1 -B KK-cycle, with Cl 1 acting by 1˝ 1 . We call this the left doubling of .E; |
| [4] | D/. |
| [5] | E˝C 2 ; D˝ 1 I 1˝ 3 / is an even unbounded A-B˝Cl 1 KK-cycle with Cl 1 acting by 1˝ 1 . We call this the right doubling of .E; |
| [6] | D/. |
| [7] | Conversely, any even A˝Cl 1 -B cycle is equivalent to the left doubling of an odd A-B cycle in KK 0 .A˝Cl 1 ; |
| [8] | B/ and any A-B˝Cl 1 cycle is the right doubling of the positive eigenspace of the non-trivial generator of Cl 1 . |
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| [27] | Walter D. van Suijlekom Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands; waltervs@math.ru.nl |
| [28] | Luuk S. Verhoeven Department of Mathematics, Middlesex College, University of Western Ontario, London, ON N6A 5B7, Canada; lverhoe@uwo.ca |
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