  This article represents a major step in the unification of the theory of
algebraic, topological and singular transition matrices by introducing a
definition which is a generalization that encompasses all of the previous
three. When this more general transition matrix satisfies the additional
requirement that it covers flow-defined Conley-index isomorphisms, one proves
algebraic and connection-existence properties. These general transition
matrices with this covering property are referred to as generalized topological
transition matrices and are used to consider connecting orbits of Morse-Smale
flows without periodic orbits, as well as those in a continuation associated to
a dynamical spectral sequence.
