  In this paper we study the topological T-dual of spaces with a non-free
circle action mainly using the stack theory method of Bunke and co-workers
\cite{Bunke1}. We first compare three formalisms for obtaining the Topological
T-dual of a semi-free $S^1$-space in a simple example. Then, we calculate the
T-dual of general KK-monopole backgrounds using the stack theory method. We
define the dyonic coordinate for these backgrounds. We introduce an approach to
Topological T-duality using classifying spaces which simultaneously generalizes
the methods of Bunke et al \cite{Bunke1} and Mathai and Wu \cite{MaWu}. Then,
we define a cohomology Gysin sequence for prinicpal bundles of stacks and
describe an application to Topological T-duality for stacks. We apply the above
to calculate the Topological T-dual of a general compact three-manifold with an
{\em arbitrary} smooth circle action. We point out a possible application of
these T-duals to higher-dimensional black holes.
