We show that Topological T-duality proposed by Mathai and Rosenberg may be used to define a T-dual for a semi-free S¹-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 S¹-space X. If B ≃ X/S¹, 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⁺ × S¹. 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 S¹-bundles E, E^# over B, T-duality gives a natural map T : H ²(E, [special characters omitted]) → H²(E#,[special characters omitted]). We define a classifying space for pairs over B consisting of a principal S^-1bundle p : X → B and a class b ∈ H²(X,[special characters omitted]). We characterize this space up to homotopy. We make a conjecture on the T-dual of an automorphism with nonzero H-flux.