The Borel regulator map on pictures. II: An example from Morse theory
Suppose that there is a smooth bundle \(E\) over an even dimensional sphere \(S^{2k}\) and a unitary representation of \(\pi_ 1 E\) with respect to which each fiber becomes acyclic. Then a fiberwise ``framed'' function on \(E\) gives a family of acyclic chain complexes over \(\mathbb{C}\) parametrized by \(S^{2k}\). By one version of algebraic \(K\)-theory this gives an element of \(K_{2k+1}\mathbb{C}\) modulo some indeterminacy. However by applying the Borel regulator map \(b_ k : K_{2k+1} \mathbb{C}\to \mathbb{R}\) one obtains a well defined real valued invariant of the bundle \(E\to S^{2k}\) which is called its higher Reidemeister torsion since it agrees with the usual Reidemeister torsion of \(E\) (actually difference between the torsion of the two components) in the case when \(k=0\). This paper gives a detailed analysis of the simplest example of this, namely, a circle bundle \(S^ 1\to E\to S^ 2\). The authors choose a specific fiberwise generalized Morse function on \(E\) and obtain a family of invertible matrices with coefficients in \(\mathbb{Z}[\pi_ 1 E]\). (The determinant of each matrix is \(1-u\) where \(u\) is the generator of \(\pi_ 1 E= \mathbb{Z}/n\).) These matrices vary by elementary row and column operations and the authors go through some lengthy technicalities in order to reduce these to column operations only. A two parameter family of invertible matrices changing by column operations gives what is called a ``picture'' and represents an element of \(K_ 3\) of the ring of coefficients. In this case the authors obtain an element \(L_ n\) of \(K_ 3 \mathbb{Z}[\xi]\) where \(\xi\) is any nontrivial \(n\)-th root of unity. Using an explicit dilogarithm formula for the Borel regulator on pictures derived in the first part of the paper (see the review above), they are able to compute the higher Reidemeister torsion invariant and they obtain \(b(L_ n)=n \text{ Im dilog}(\xi)\). Bismut and Lott have given an analytic definition of higher Reidemeister torsion and computed it in many cases. For circle bundles over \(S^ 2\) their calculation also gives \(n\text{ Im dilog}(\xi)\). It is not known whether the two definitions of higher Reidemeister torsion agree in other cases.
scientific article; zbMATH DE number 488350