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On the compatibility between the differential topological index and the analytic Bunke-Schick push-forward construction. (English) Zbl 1427.19005
Given a proper submersion $$\pi : X \to Y$$ for which the vertical tangent bundle of $$X$$ has a Spin$$^c$$-structure, the analytic push-forward in Bunke-Schick’s differential $$K$$-theory [U. Bunke and T. Schick, Astérisque 328, 45–135 (2009; Zbl 1202.19007)] can be considered an analytic index. On the other hand, there is also a topological index for Freed-Lott’s differential $$K$$-theory [D. S. Freed and J. Lott, Geom. Topol. 14, No. 2, 903–966 (2010; Zbl 1197.58007)]. These two theories are isomorphic and the author introduces a modification of this topological index which is compatible with the analytic push-forward, in the sense that the diagram consisting of these isomorphisms and indices commutes. Alternatively, this can be interpreted as the construction of a topological index for Bunke-Schick’s differential $$K$$-theory which agrees with the analytic index. In particular, this new index only depends on the smooth $$K$$-orientation of $$\pi$$ and it is functorial under composition of smooth $$K$$-oriented proper submersions.
The paper contains an elementary proof of a Riemann-Roch type theorem for Lott’s $$\mathbb{R}/\mathbb{Z}$$-$$K$$-theory using the previous result. Other consequences include the invariance of Lott’s analytic index in $$\mathbb{R}/\mathbb{Z}$$-$$K$$-theory under bordism, a geometric realization of higher Adams $$e$$-invariants for families of framed manifolds and an index theorem for geometric families of $$\mathbb{Z}/k$$-manifolds, in particular, an index theorem for flat bundles over manifolds with finite fundamental groups.
The compatibility of the two indices is shown by first proving that the difference between the two compositions in the diagram comes from an element in $$\mathbb{R}/\mathbb{Z}$$-$$K$$-theory. This element will vanish if and only if it vanishes after pairing it with any odd geometric $$K$$-homology class. These pairings equal differences of reduced eta-invariants and the author uses several properties and reductions to establish the vanishing of these differences.
##### MSC:
 19L50 Twisted $$K$$-theory; differential $$K$$-theory 58J20 Index theory and related fixed-point theorems on manifolds
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