  We consider a generalization of the axioms of a TQFT, so called
half-projective TQFT's, with an anomaly, $x^{\mu}$, in the composition law.
$\mu$ is a coboundary on the cobordism categories with non-negative, integer
values. The element $x$ of the ring over which the TQFT is defined does not
have to be invertible. In particular, it may be 0. This modification makes it
possible to extend quantum-invariants, which vanish on $S^1\times S^2$, to
non-trivial TQFT's. (A TQFT in the sense of Atiyah with this property has to be
trivial all together). Under a few natural assumptions the notion of a
half-projective TQFT is shown to be the only possible generalization. Based on
separate work with Lyubashenko on connected TQFT's, we construct a large class
of half-projective TQFT's with $x=0$. Their invariants vanish on $S^1\times
S^2$, and they coincide with the Hennings invariant for non-semisimple Hopf
algebras. Several toplogical tools that are relevant for vanishing properties
of such TQFT's are developed. They are concerned with connectivity properties
of cobordisms, as for example maximal non-separating surfaces. We introduce in
particular the notions of ``interior'' homotopy and homology groups, and of
coordinate graphs, which are functions on cobordisms with values in the
morphisms of a graph category. For applications we will prove that
half-projective TQFT's with $x=0$ vanish on cobordisms with infinite interior
homology, and we argue that the order of divergence of the TQFT on a cobordism
in the ``classical limit'' can be estimated by the rank of its maximal free
interior group.
