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Advances in Noncommutative Geometry pp 691–751Cite as

Higher invariants in noncommutative geometry

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Abstract

We give a survey on higher invariants in noncommutative geometry and their applications to differential geometry and topology.

Dedicated to Alain Connes with great admiration

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Notes

  1. 1.

    In other words, there is no need to pass to the operator \(\mathcal {F}\) or \(\mathcal {F}(t)\) as in the general case.

  2. 2.

    Note that in this case, all finite order elements in come from Γ.

  3. 3.

    Proposition 6.14 first appeared in [WY]. The original statement in [WY] seems to contain a minor error when d is even, the version we state in this survey and its proof can be found in [XYZ].

  4. 4.

    Without loss of generality, we can assume N R(Z) is Γ-invariant.

  5. 5.

    A geometric construction of the Siebenmann periodicity map was given by Cappell and Weinberger [CW].

  6. 6.

    Let us review how the homomorphism \(\psi _\ast : K_{1}(C_{L,0}^\ast (\widetilde M)^\Gamma )\to K_{1}(C_{L,0}^\ast (\widetilde M)^\Gamma )\) is defined. The map ψ: M → M lifts to a map \(\widetilde \psi \colon \widetilde M \to \widetilde M\). However, to view \(\widetilde \psi \) as a Γ-equivariant map, we need to use two different actions of Γ on \(\widetilde M\). Let τ be a right action of Γ on \(\widetilde M\) through deck transformations. Then we define a new action τ′ of Γ on \(\widetilde M\) by \(\tau ^{\prime }_g = \tau _{\psi _\ast (g)}\), where ψ ∗:  Γ → Γ is the automorphism induced by ψ. It is easy to see that \(\widetilde \psi \colon \widetilde M \to \widetilde M\) is Γ-equivariant, when Γ acts on the first copy of \(\widetilde M\) by τ and the second copy of \(\widetilde M\) by τ′. Let us denote the corresponding C ∗-algebras by \(C_{L,0}^\ast (\widetilde M)_{\tau }^\Gamma \) and \(C_{L,0}^\ast (\widetilde M)_{\tau '}^\Gamma \). Observe that, despite the two different actions of Γ on \(\widetilde M\), the two C ∗-algebras \(C_{L,0}^\ast (\widetilde M)_{\tau }^\Gamma \) and \( C_{L,0}^\ast (\widetilde M)_{\tau '}^\Gamma \) are canonically identical, since an operator is invariant under the action τ if and only if it is invariant under the action τ′.

  7. 7.

    Precisely speaking, φ only defines an outer automorphism of Γ, and one needs to make a specific choice of a representative in Aut( Γ). In any case, any such choice will work for the proof.

  8. 8.

    The C ∗-algebra \(C_{L,0}^\ast (E\Gamma )^\Gamma \) is the inductive limit of \(C_{L,0}^\ast (Y)^\Gamma \), where Y  ranges over all Γ-cocompact subspaces of E Γ.

  9. 9.

    Here “invertible” means being invertible on the universal cover of the manifold.

  10. 10.

    We refer the reader to [CWXY] for details on how to identify the formula for \(\eta _{\varphi }(\widetilde D)\) in Theorem 10.7 with the periodic version of Lott’s noncommutative-differential higher eta invariant.

  11. 11.

    In fact, even more is true. One can use the same techniques developed in [CWXY] to show that if \(\mathcal A\) is smooth dense subalgebra of \(C_r^\ast (\Gamma )\) for any group Γ (not necessarily hyperbolic) and in addition \(\mathcal A\) is a Fréchet locally m-convex algebra, then there is a well-defined delocalized Connes–Chern character \(Ch_{deloc}\colon K_i(C^*_{L,0}(\widetilde M)^\Gamma )\to \overline {HC}^{deloc}_{\ast }(\mathcal A)\). Of course, in order to pair such a delocalized Connes–Chern character with a cyclic cocycle of \(\mathbb C\Gamma \), the key remaining challenge is to continuously extend this cyclic cocycle of \(\mathbb C\Gamma \) to a cyclic cocycle of \(\mathcal A\).

  12. 12.

    Here the definition of cyclic homology of \(\mathcal B\) takes the topology of \(\mathcal B\) into account, cf. [C2, Section II.5].

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Acknowledgements

The authors wish to thank Alain Connes for numerous inspiring discussions. The author “Zhizhang Xie” is partially supported by NSF 1500823, NSF 1800737. The author “Guoliang Yu” is partially supported by NSF 1700021, NSF 1564398.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Zhizhang Xie

  2. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Guoliang Yu

  3. Shanghai Center for Mathematical Sciences, Shanghai, China

    Guoliang Yu

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  1. Zhizhang Xie
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Correspondence to Guoliang Yu .

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  1. Department of Physics Institute for Mathematics, Astrophysics and Particle Physics, Nijmegen, The Netherlands

    Ali Chamseddine

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA

    Caterina Consani

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Nigel Higson

  4. Department of Mathematics, The University of Western Ontario, London, ON, Canada

    Masoud Khalkhali

  5. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Henri Moscovici

  6. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Guoliang Yu

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Xie, Z., Yu, G. (2019). Higher invariants in noncommutative geometry. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_12

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