LAUSR

Abstract In Bouwknegt et al. (2015) [3, 4], we introduced spherical T-duality, which relates pairs of the form ( P , H ) consisting of an oriented S 3 -bundle P → M and a 7-cocycle H on P called the 7-flux. Intuitively, the spherical T-dual is another such pair ( P ˆ , H ˆ ) and spherical T-duality exchanges the 7-flux with the Euler class, upon fixing the Pontryagin class and the second Stiefel–Whitney class. Unless dim ( M ) ≤ 4 , not all pairs admit spherical T-duals and the spherical T-duals are not always unique. In this paper, we define a canonical Poincare virtual line bundle P on S 3 × S 3 (actually also for S n × S n ) and the spherical Fourier–Mukai transform, which implements a degree shifting isomorphism in K-theory on the trivial S 3 -bundle. This is then used to prove that all spherical T-dualities induce natural degree-shifting isomorphisms between the 7-twisted K-theories of the pairs ( P , H ) and ( P ˆ , H ˆ ) when dim ( M ) ≤ 4 , improving our earlier results.