  New algebraic relations are presented, involving anticommuting Grassmann
variables and Berezin integral, and corresponding naturally to Pachner moves in
three and four dimensions. These relations have been found experimentally -
using symbolic computer calculations; their essential new feature is that,
although they can be treated as deformations of relations corresponding to
torsions of acyclic complexes, they can no longer be explained in such terms.
In the simpler case of three dimensions, we define an invariant, based on our
relations, of a piecewise-linear manifold with triangulated boundary, and
present example calculations confirming its nontriviality.
