  Given a fibration over the circle, we relate the eigenspace decomposition of
the algebraic monodromy, the homological finiteness properties of the fiber,
and the formality properties of the total space. In the process, we prove a
more general result about iterated group extensions. As an application, we
obtain new criteria for formality of spaces, and 1-formality of groups,
illustrated by bundle constructions and various examples from low-dimensional
topology and singularity theory.
