but where the blowup locus is a product of $T$ and the symplectic mapping torus of a certain automorphism of $K \times K$ (embedded into $K^7$ by using the $h$-principle). It is known that the symplectic automorphism group has many connected components which are not detected by classical topological means (see for instance [@seidel04b], or [@seidel-thomas99] for a mirror symmetry viewpoint). Based on that, one can ensure that $B^{\mathit{triv}}$ is diffeomorphic to $B$, and that their symplectic structures are deformation equivalent. Nevertheless, for a specific choice of automorphism, we will show:

> *$B$ and $B^{\mathit{triv}}$ are not symplectically isomorphic.*

The construction of these manifolds is similar to that of the first known examples of distinct but deformation equivalent symplectic structures [@mcduff87], which were also based on blowing up. That paper used (roughly speaking) a bordism-valued refinement of Gromov-Witten theory as an invariant. Because such refinements are hard to define and compute, we can’t say how they would behave in our situation, but intuitively it seems unlikely that they would be able to distinguish the two manifolds above. In any case, the approach taken in this paper is quite different.

[**The invariant.**]{} Let’s temporarily go back to the simpler case of 