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Flat vector bundles, direct images and higher real analytic torsion. (English) Zbl 0837.58028
The purpose of the paper is to extend the Ray-Singer analytic torsion from an invariant of a smooth manifold to an invariant of a smooth parametrized family of manifolds. Let \[ Z\to M@>\pi>> B \] be a fiber bundle with closed fibers \(Z_b= \pi^{-1}(b)\). To a flat complex vector bundle \(F\) on \(M\) one can associate certain characteristic classes \(c_k(F)\in H^k(M; \mathbb{R})\) [see J. L. Dupont, Topology 15, 233-245 (1976; Zbl 0331.55012)]. Let \(e(TZ)\in H^{\dim(Z)}(M; \mathbb{R})\) denote the Euler class of the vertical tangent bundle of the fibration. Let \(H^p(Z, F|_Z)\) be the flat complex vector bundle on \(B\) whose fiber over \(b\in B\) is isomorphic to \(H^p(Z_b, F|_{Z_b})\). The following smooth analog of the Riemann-Roch-Grothendieck theorem is shown:
Theorem. For any positive odd integer \(k\), \[ \sum^{\dim(Z)}_{p= 0} (- 1)^p c_k(H^p(Z, F|_Z))= \int_Z e(TZ) c_k(F)\in H^k(B; \mathbb{R}). \] This follows from a more refined differential form version. On the level of differential forms the difference of the left- hand side and the right-hand side is shown to equal the exterior derivative of the so-called “higher analytic torsion form”. This is a certain differential form on \(B\) of mixed even degree whose degree-zero part is the function which assigns to each \(b\in B\) the Ray-Singer torsion of \((Z_b, F|_{Z_b})\).
The results of this paper have been announced by the authors in C. R. Acad. Sci., Paris, Sér. I 316, No. 5, 477-482 (1993; Zbl 0780.57023).
Reviewer: C.Bär (Freiburg)

MSC:
58J20 Index theory and related fixed-point theorems on manifolds
57R57 Applications of global analysis to structures on manifolds
57R19 Algebraic topology on manifolds and differential topology
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