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Transfer index conjecture for differential algebraic K-theory

Let π:M→B\pi:M\to B be a proper submersion between manifolds, let VV be a bundle of RR-modules with geometry hVh^{V}, let gπg^{\pi} be a Riemannian structure on π\pi, and let hL2Riπ∗(V)h_{L^{2}}^{R^{i}\pi_{*}(V)} be the fibrewise Hodge-theoretic geometry on Riπ∗(V)R^{i}\pi_{*}(V). Write cycl⁡^\hat{\operatorname{cycl}} for the differential algebraic KK-theory cycle map, tr⁡^\hat{\operatorname{tr}} for the differential Becker--Gottlieb transfer, and aa for the map from differential forms to differential cohomology. Then the transfer index conjecture (TIC).

∑i≥0(−1)icycl⁡^(Riπ∗(V),hL2Riπ∗(V))+a(T(π,gπ,V,hV))=tr⁡^(cycl⁡^(V,hV))\sum_{i\ge 0}(-1)^{i}\hat{\operatorname{cycl}}(R^{i}\pi_{*}(V),h_{L^{2}}^{R^{i}\pi_{*}(V)})+a({\mathcal{T}}(\pi,g^{\pi},V,h^{V}))=\hat{\operatorname{tr}}(\hat{\operatorname{cycl}}(V,h^{V}))

in KR^0(B)\widehat{KR}^{0}(B). This is the differential refinement of the index identity comparing the sheaf-theoretic push-forward with the Becker--Gottlieb transfer; the correction term is given by the higher analytic torsion form. The underlying equality in algebraic KK-theory follows from the Dwyer--Weiss--Williams index theorem, while the regulator image is governed by the Bismut--Lott index theorem; the differential refinement is the conjectural assertion.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Transfer index conjecture for differential algebraic K-theory

    Let (π:W→B,g)(\pi:W\to B,g) be a proper submersion with Riemannian structure, and let (V,hV)(\mathcal V,h^{\mathcal V}) be a locally constant sheaf of finitely generated RR-modules with geometry. The topological index is the differential transfer of its cycle class, and the analytic index is the alternating sum of the cycle classes of the higher direct images, equipped with their Hodge-theoretic geometries, together with the Bismut–Lott higher-torsion correction. These define maps of \Bundlegeomδ\Bundle^{\delta}_{geom}-sets

    \hindtop,\hindan:\blocgeomd⟶KR^0r.\hind^{top},\hind^{an}:\bloc_{geom}d\longrightarrow \widehat{KR}^{0}r.

    Transfer index conjecture. The two maps of \Bundlegeomδ\Bundle^{\delta}_{geom}-sets are equal:

    \hindtop=\hindan:\blocgeomd→KR^0r.\hind^{top}=\hind^{an}:\bloc_{geom}d\to \widehat{KR}^{0}r.

    The equality is known after applying the curvature map and the underlying algebraic KK-theory map, by the Bismut–Lott and Dwyer–Weiss–Williams index theorems respectively. The conjecture asserts equality in differential algebraic KK-theory itself, with the remaining difference factoring through a natural transformation valued in the appropriate quotient of HA−1HA^{-1}; its vanishing is open in the source.

    source: Ulrich Bunke and David Gepner, “Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory”, arXiv:1306.0247 (2016).

Sources & referencesView supporting material

Primary source

Ulrich Bunke and Georg Tamme, “Regulators and cycle maps in higher-dimensional differential algebraic K-theory”, arXiv:1209.6451 (2015).

algebraic-k-theoryanalytic-torsiondifferential-cohomologyindex-theory

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