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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.

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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