  On an oriented, compact, connected, real four-dimensional manifold, $M$, we
introduce a topological Lagrangian gauge field theory with a Bogomol'nyi
structure that leads to non-singular, finite-Action, stable solutions to the
variational field equations. These soliton-like solutions are analogous to the
instanton in Yang-Mills theory. Unlike Yang-Mills instantons, however,
`topological' instantons are independent of any underlying metric structure,
and, in particular, they are independent of the metric signature. We show that
when the topology of the underlying manifold, $M$, is equipped with a complex
K\"ahler structure, and $M$ is interpreted as space-time, then the moduli space
of topological instantons---the space of motions---is a finite-dimensional,
smooth, Hausdorff manifold with a natural symplectic structure. We identify
space-time topologies which lead to the physical stability of topological
instanton field configurations compatible with the additional geometric
structures. The spaces of motion for $U(1)$ topological instantons over either
minimal elliptic or algebraic complex space-times with irregularity $q=2$ are
examined.
