  We construct examples in any odd dimension of contact manifolds with finite
and non-zero algebraic torsion (in the sense of Latschev-Wendl), which are
therefore tight and do not admit strong symplectic fillings. We prove that
Giroux torsion implies algebraic $1$-torsion in any odd dimension, which proves
a conjecture by Massot-Niederkrueger-Wendl. 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 SFT machinery. We give a tentative definition of a
higher-dimensional spinal open book decomposition, based on the $3$-dimensional
one of Lisi-van Horn Morris-Wendl. An appendix written in co-authorship with
Richard Siefring gives a basic outline of the intersection theory for punctured
holomorphic curves and hypersurfaces, which generalizes his $3$-dimensional
results of Siefring to higher dimensions. From the intersection theory we
obtain an application to codimension-$2$ holomorphic foliations, which we use
to restrict the behaviour of holomorphic curves in our examples.
