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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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