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Higher Franz-Reidemeister torsion. (English) Zbl 1016.19001
AMS/IP Studies in Advanced Mathematics. 31. Providence, RI: American Mathematical Society (AMS). Somerville, MA: International Press, xxii, 370 p. (2002).
The bulk of this book is devoted to developing the definition of higher Franz-Reidemeister torsion classes \(\tau_k(E;\mathcal F)\in H^{2k}(B;\mathbb R)\) when \(p:E\to B\) is a smooth manifold bundle with compact manifold fiber \(M\) so that \(\pi_1(B)\) acts trivially on the homology groups of the fiber \(H_*(M,\mathcal F)\), where \(\mathcal F\) is a Hermitian coefficient system for \(E\). This generalizes the construction of higher Franz-Reidemeister torsion invariants in the case of pseudoisotopies, where one gets them from higher Whitehead torsion through Waldhausen’s \(K\)-theory space \(A(X)\) and its relationship to the algebraic \(K\)-theory of discrete rings.
In the present context one first uses Morse theory in the form of the framed function theorem of the author to go from the smooth bundle to a mapping from the base \(B\) into a Whitehead space, built using Volodin \(K\)-theory. Then one composes with a Borel regulator map.
Explicit computations are given for oriented linear circle bundles, oriented linear sphere bundles, lens spaces bundles, nonlinear disc bundles, the mapping class group and the Torelli group. Some familiarity with earlier work is needed to follow the text.

19-02 Research exposition (monographs, survey articles) pertaining to \(K\)-theory
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
57R45 Singularities of differentiable mappings in differential topology
19F27 √Čtale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
19D10 Algebraic \(K\)-theory of spaces
55R40 Homology of classifying spaces and characteristic classes in algebraic topology