A bilinear form of dilogarithm and motivic regulator map

The Bloch–Wigner function D2 is a single-valued version of a dilogarithm function and is used by Bloch to describe the Borel regulator map from K3(C) into R explicitly (c.f. [Bloch, Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves, American Mathematical Society, Providence, RI, 2000]). We introduce a new way to formulate a single-valued dilogarithm function and use it to explicitly define a motivic regulator map for , defined in terms of the motivic complex of Goodwillie and Lichtenbaum. We also detect certain explicit nonzero elements in the motivic cohomology group. Throughout this paper, a path will be a C1-function from the unit interval [0,1] into C-{0}.