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

We relate spherical T-duality for oriented linear $\mathrm{S}^3$-bundles over $\mathrm{S}^4$ (the Milnor bundles $M_{m,n}$, which are $\mathrm{S}^3$-principal exactly when $m=0$ or $n=0$, and whose total spaces are homotopy $7$-spheres exactly when $m+n=\pm1$) to $\star$-diagrams and to a higher-dimensional generalization of the logarithmic transformations of $4$-manifold topology. For an $\mathrm{S}^3$-principal pair $(P,H)$, $(\widehat P,\widehat H)$ over $\mathrm{S}^4$, we show that the T-duality correspondence space $P\times_{\mathrm{S}^4}\widehat P$ is itself a $\star$-diagram of a distinguished type, which we call \emph{bifree}, and that bifree $\star$-diagrams are precisely the fiber products of principal bundles; spherical T-duality of the decorated pair is then a condition on the fluxes carried by that diagram. For bundles of equal Euler class $ku$, principal or not, we show that the two bundles are spherical T-dual with the diagonal fluxes $[k]$, and that they occur as the two base manifolds of an explicit $\star$-diagram, obtained by pulling back a principal Milnor bundle; this diagram is never bifree. We then introduce product-preserving generalized logarithmic transformations on products $\Sigma\times\mathrm{S}^1$ of homotopy spheres with the circle, and prove that, after stabilization by $\mathrm{S}^1$, the spherical T-dualities between homotopy $7$-spheres are realized by such transformations. In particular, $\Sigma^7_{GM}\times\mathrm{S}^1$ is obtained from $\mathrm{S}^7\times\mathrm{S}^1$ by one of them, where $\Sigma^7_{GM}$ denotes the Gromoll--Meyer exotic sphere: spherical T-duality relates distinct smooth structures on the topological $7$-sphere, and the relation is implemented by an explicit cut-and-paste operation.