Topological T-duality, Automorphisms and Classifying Spaces

We extend the formalism of Topological T-duality to spaces which are the total space of a principal $S^1$-bundle $p:E \to W$ with an $H$-flux in $H^3(E,Z)$ together the together with an automorphism of the continuous-trace algebra on $E$ determined by $H$. The automorphism is a `topological approximation' to a gerby gauge transformation of spacetime. We motivate this physically from Buscher's Rules for T-duality. Using the Equivariant Brauer Group, we connect this problem to the $C^{\ast}$-algebraic formalism of Topological T-duality of Mathai and Rosenberg. We show that the study of this problem leads to the study of a purely topological problem, namely, Topological T-duality of triples $(p,b,H)$ consisting of isomorphism classes of a principal circle bundle $p:X \to B$ and classes $b \in H^2(X,Z)$ and $H \in H^3(X,Z).$ We construct a classifying space $R_{3,2}$ for triples in a manner similar to the work of Bunke and Schick \cite{Bunke}. We characterize $R_{3,2}$ up to homotopy and study some of its properties. We show that it possesses a natural self-map which induces T-duality for triples. We study some properties of this map.