Configuration complexes and a variant of Cathelineau's complex in weight 3
Abstract
In this paper we consider the Grassmannian complex of projective configurations in weight 2 and 3, and Cathelineau's infinitesimal polylogarithmic complexes. Our main result is a morphism of complexes between the Grassmannian complex and the associated infinitesimal polylogarithmic complex. In order to establish this connection we introduce an
F
-vector space
β
D
2
(F)
, which is an intermediate structure between a $\varmathbb{Z}$-module
B
2
(F)
(scissors congruence group for
F
) and Cathelineau's
F
-vector space
β
2
(F)
which is an infinitesimal version of it. The structure of
β
D
2
(F)
is also infinitesimal but it has the advantage of satisfying similar functional equations as the group
B
2
(F)
. We put this in a complex to form a variant of Cathelineau's infinitesimal complex for weight 2.