The Borel regulator map on pictures. I: A dilogarithm formula
For any odd \(k\) the Borel regulator map is a map \(K_ k\mathbb{C}\to \mathbb{R}\) given by composing the Hurewicz map \(K_ k \mathbb{C}\to H_ k GL(\mathbb{C})\) with the map \(b_ *: H_ k GL(\mathbb{C})\to \mathbb{R}\) induced by an essentially unique continuous cohomology class \(b\in H_{\text{cont}}^ k (GL(\mathbb{C}),\mathbb{R})\). In the case \(k=3\) this cohomology class is the only one whose restriction to \(SL_ 2(\mathbb{C})\) is represented by the 3-cocycle given by \(f(A,B,C)= D_ 2 r(e_ 1,Ae_ 1,Be_ 1,Ce_ 1)\) where \(e_ 1=(1,0)^ t\) and \(r\) is the cross ratio specified by \(r(\infty,0,1,z)=z\) and \(D_ 2: P^ 1(\mathbb{C})\to \mathbb{R}\) is the Bloch-Wigner dilogarithm function \[ D_ 2(z)= \arg(1-z) \log | z|+ \text{Im} \sum_{n=1}^ \infty {{z^ n} \over {n^ 2}}. \] Elements of \(K_ 3\mathbb{C}\) are represented by ``pictures''. These are planar diagrams with regions labeled by invertible complex matrices which change by elementary operations along the edges. The generator of \(K_ 3 \mathbb{C}\approx \mathbb{Z}/48\) is given as a picture in the paper. The formula for evaluating the Borel regulator on a picture for \(K_ 3\mathbb{C}\) is roughly as follows. We first dualize the picture and triangulate the plane so that each vertex has an invertible matrix attached to it and each edge has an elementary operation. For a triangle with vertices \(R_ 1,R_ 2,R_ 3\in GL_ n(\mathbb{C})\) we take a maximal flag \(F\) in \(\mathbb{C}^ n\) and consider \((F,R_ 1 F,R_ 2 F,R_ 3 F)\). Take all possible configurations of four points on a projective line by intersecting and restricting these four flags, apply the function \(D_ 2 r\), take the sum and average over all flags. In part II of this paper (Igusa and Klein; see the following review) the formula is simplified and nontrivial examples are explicitly computed.
scientific article; zbMATH DE number 488349