A Geometric Realization of Spherical T-Duality via $\star$-Diagrams

This paper establishes an equivalence between two distinct frameworks for constructing and relating smooth manifolds: the geometric theory of \emph{$\star$-diagrams} and the string-theory-inspired notion of \emph{spherical T-duality}. We prove that for linear ${\mathrm{S}}^3$-bundles over the 4-sphere, the existence of a $\star$-diagram connecting two such bundles is equivalent to them forming a spherical T-dual pair. This result provides a concrete geometric realization of spherical T-duality, interpreting its abstract cohomological definitions in the language of differential geometry. To forge this connection, we introduce a higher-dimensional generalization of \emph{logarithmic transformations}. These topological surgeries change the diffeomorphism type of the homology $\mathrm{\Sigma }\times {\mathrm{S}}^1$, where $\mathrm{\Sigma }$ is a homotopy sphere. Forgetting the ${\mathrm{S}}^1$-factor, they realize the constructed spherical T-dualities. Furthermore, we show that the known isomorphisms in the equivariant K-theory and cohomology between the T-dual manifolds are a direct consequence of an underlying \emph{Morita equivalence} between the action groupoids naturally associated with the base manifolds in a $\star$-diagram.