  We construct infinitely many non-diffeomorphic examples of $5$-dimensional
contact manifolds which are tight, admit no strong fillings, and do not have
Giroux torsion. We obtain obstruction results for symplectic cobordisms, for
which we give a proof not relying on the polyfold abstract perturbation scheme
for SFT. These results are part of the author's PhD thesis, and are the first
applications of higher-dimensional Siefring intersection theory for holomorphic
curves and hypersurfaces.
