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$$\mathbb{R} /\mathbb{Z}$$ index theory. (English) Zbl 0840.58044
Let $$Z\to M\to B$$ be a smooth fiber bundle whose fiber $$Z$$ is a closed even-dimensional manifold and whose base $$B$$ is a compact manifold; moreover, the vertical tangent bundle $$TZ$$ is endowed with a spin-structure. Let $$L_Z$$ be the Hermitian line bundle on $$M$$ which is associated to this spin-structure on $$TZ$$.
The aim of this paper is to introduce an index theory in which the indices take values in $$\mathbb{R}/\mathbb{Z}$$ in the sense that it is based on a generalized cohomology theory whose even coefficient groups are copies of $$\mathbb{R}/\mathbb{Z}$$. To describe this cohomology theory, the notion of $$K^*_{\mathbb{C}/\mathbb{Z}}(M)$$ – the $$K$$-theory of a manifold $$M$$ with $$\mathbb{C}/\mathbb{Z}$$ coefficients – and Karoubi’s geometric description of $$K^{-1}_{\mathbb{C}/\mathbb{Z}}(M)$$ are used [see M. Karoubi, ‘Homologie cyclique et $$K$$-théorie’ (1987; Zbl 0648.18008) and $$K$$-Theory 4, No. 1, 55-87 (1990; Zbl 0716.57018)].
By adding Hermitian structures on the vector bundles, the author obtains a geometric description of $$K^{- 1}_{\mathbb{R}/\mathbb{Z}}(M)$$ – the $$K$$-theory of $$M$$ with $$\mathbb{R}/\mathbb{Z}$$ coefficients. So, in such a manner, a map $$\text{ind}_{\text{top}}: K^{- 1}_{\mathbb{R}/\mathbb{Z}}(M)\to K^{- 1}_{\mathbb{R}/\mathbb{Z}}(B)$$ is defined by topological means.
The major point of this paper is the construction of a corresponding analytic index map. Namely, given a cocycle $$\mathcal E$$ for $$K^{- 1}_{\mathbb{R}/\mathbb{Z}}(M)$$, the author first defines an analytic index $$\text{ind}_{\text{an}}({\mathcal E})\in K^{- 1}_{\mathbb{R}/\mathbb{Z}}(B)$$ when $$\mathcal E$$ satisfies a certain technical assumption. To define this index, the vertical tangent bundle $$TZ$$ is endowed with a metric and $$L_Z$$ with a Hermitian connection. The technical assumption is that the kernels of the fiberwise Dirac-type operators form a vector bundle on $$B$$. The construction of $$\text{ind}_{\text{an}}({\mathcal E})$$ involves this vector bundle on $$B$$ and the eta-form of Bismut and Cheeger [see J. M. Bismut and J. Cheeger, J. Am. Math. Soc. 2, No. 1, 33-70 (1989; Zbl 0671.58037) and X. Dai, J. Am. Math. Soc. 4, No. 2, 265-321 (1991; Zbl 0736.58039)].
If $$\mathcal E$$ does not satisfy the technical assumption, the author effectively deforms it to a cocycle which does, and again defines $$\text{ind}_{\text{an}}({\mathcal E})$$.
The main result is: $\text{ind}_{\text{an}}({\mathcal E})= \text{ind}_{\text{top}}({\mathcal E}).$ The method of proof is to show that there exists an equality after pairing both sides of the equation with an arbitrary element of the odd-dimensional $$K$$-homology of $$B$$. These pairings are given by eta-invariants, and the main technical feature of the proof is the computation of adiabatic limits of eta-invariants.

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 46L05 General theory of $$C^*$$-algebras 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 19K56 Index theory
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