##### Abstract

We show that Topological T-duality proposed by Mathai and Rosenberg may
be used to define a T-dual for a semi-free S^1-space. In particular,
we argue that it gives the physical T-dual for a system of
n Kaluza-Klein (KK) monopoles.
We show that the `dyonic coordinates' well known in the physics literature
may be incorporated within this formalism of Topological T-duality.
We study some formal properties of topological T-duality: We note
that Topological T-duality naturally defines a T-dual of any semi-free
S^1-space X. If B \simeq X/S^1, X is naturally associated
to a Hitchin 2-gerbe on B^{+}. We also note that T-duals of
such spaces may be naturally associated to Hitchin 3-gerbes on
B^{+} \times S^1. We demonstrate that Topological T-duality gives a natural
mapping between these two gerbes.
We use the Equivariant Brauer Group to model a space with a
B-field or a H-flux. We note that each step of the natural filtration on
this group corresponds to one of the gauge fields of the H-flux.
We note that given a T-dual pair of principal S^1-bundles
E,E^{\#} over B, T-duality gives a natural map
T:H^2(E,\KZ) \to H^2(E^{\#},\KZ).
We define a classifying space for pairs over B consisting of a principal
S^-1bundle p:X \to B and a class b is an element of H^2(X,\KZ).
We characterize this space up to homotopy.
We make a conjecture on the T-dual of an automorphism with nonzero $H-$flux.