  We exploit a uniform recursive procedure using preferred contractions of
targets $C_*$ to construct morphisms $B_* \to C_*$ between chain complexes in a
wide variety of situations. Examples include classical Alexander-Whitney and
Eilenberg-Zilber maps, chain maps related to homology and cohomology
operations, operad structure maps for various chain complex operads, and chain
maps that define morphisms between operads. The procedure includes functorial
chain maps $F_*(X_i) \to K_*(X_i)$, using models for generators of the domain
and contractions of $K_*$ applied to these models. Various uniqueness theorems
characterize the chain maps produced by our procedures. We give unified
extended treatments of the operads known as the Barratt-Eccles operad and the
surjection operads. In a subsequent paper we plan to use the results and
methods of this paper to establish properties of the Steenrod algebra of mod p
cohomology operations at the cochain level. Classical treatments of Steenrod
operations used less explicit acyclic model methods to deduce properties of
operations at the cohomology level. Our paper is comprehensive and includes
many examples and specific results of a somewhat technical nature. But the main
point is that a rather easily described explicit procedure clarifies aspects of
chain maps defined only up to homotopy that were important in the development
of cohomology operations roughly seventy years ago, as well as complicated
chain maps related to operad structure maps and operad morphisms developed in a
rather ad hoc manner roughly twenty five years ago. Sections 0, 1, and 2
constitute an overview and summary of the paper.
