  This is the second in a series of papers where we construct an invariant of a
four-dimensional piecewise linear manifold $M$ with a given middle cohomology
class $h\in H^2(M,\mathbb C)$. This invariant is the square root of the torsion
of unusual chain complex introduced in Part I (arXiv:1605.06498) of our work,
multiplied by a correcting factor. Here we find this factor by studying the
behavior of our construction under all four-dimensional Pachner moves, and show
that it can be represented in a multiplicative form: a product of same-type
multipliers over all 2-faces, multiplied by a product of same-type multipliers
over all pentachora.
