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Index theory in spaces of manifolds

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Abstract

We formulate and prove a generalization of the Atiyah-Singer family index theorem in the context of the theory of spaces of manifolds à la Madsen, Tillmann, Weiss, Galatius and Randal-Williams. Our results are for Dirac-type operators linear over arbitrary \(C^*\)-algebras.

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Notes

  1. We denote the K-theory spectrum of a Real graded \(\mathrm {C}^{*}\)-algebra \(\mathbf {A}\) by \(\mathbb {K}(\mathbf {A})\) and the nth space in this spectrum by \(\mathbb {K}(\mathbf {A})_n\). Hence \([X; \mathbb {K}(\mathbf {Cl}^{d,0})_n] \cong KO^{n-d}(X)\).

  2. We denote the pullback along inclusions by the restriction symbol.

  3. Everything in this paper can easily be “complexified”, by ignoring the Real structure at every place.

  4. As explained in [8, Remark 3.5], we take a Grothendieck universe and consider all cycles which are contained in this universe.

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  1. Mathematisches Institut, Universität Münster, Einsteinstraße 62, 48149, Münster, Germany

    Johannes Ebert

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  1. Johannes Ebert
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Correspondence to Johannes Ebert.

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Communicated by Thomas Schick.

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Partially supported by the SFB 878 “Groups, Geometry and Actions”.

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Ebert, J. Index theory in spaces of manifolds. Math. Ann. 374, 931–962 (2019). https://doi.org/10.1007/s00208-019-01809-4

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