  We study the vector spaces and integer lattices of cuts and flows associated
with an arbitrary finite CW complex, and their relationships to group
invariants including the critical group of a complex. Our results extend to
higher dimension the theory of cuts and flows in graphs, most notably the work
of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut
and flow spaces, interpret their coefficients topologically, and give
sufficient conditions for them to be integral bases of the cut and flow
lattices. Second, we determine the precise relationships between the
discriminant groups of the cut and flow lattices and the higher critical and
cocritical groups with error terms corresponding to torsion (co)homology. As an
application, we generalize a result of Kotani and Sunada to give bounds for the
complexity, girth, and connectivity of a complex in terms of Hermite's
constant.
