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

Abstract

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 $m+n=\pm 1$) to $\star$-diagrams and to a higher-dimensional generalization of the logarithmic transformations of $4$-manifold topology. For an ${\mathrm{S}}^3$-principal pair $(P,H)$, $(\hat{P},\hat{H})$ over ${\mathrm{S}}^4$, we show that the T-duality correspondence space $P{\times }_{{\mathrm{S}}^4}\hat{P}$ 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, ${\Sigma }_{GM}^7\times {\mathrm{S}}^1$ is obtained from ${\mathrm{S}}^7\times {\mathrm{S}}^1$ by one of them, where ${\Sigma }_{GM}^7$ 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.

Accepted version. To appear in Advances in Mathematics