  Gluing two manifolds M_1 and M_2 with a common boundary S yields a closed
manifold M. Extending to formal linear combinations x=Sum_i(a_i M_i) yields a
sesquilinear pairing p=<,> with values in (formal linear combinations of)
closed manifolds. Topological quantum field theory (TQFT) represents this
universal pairing p onto a finite dimensional quotient pairing q with values in
C which in physically motivated cases is positive definite. To see if such a
"unitary" TQFT can potentially detect any nontrivial x, we ask if <x,x> is
non-zero whenever x is non-zero. If this is the case, we call the pairing p
positive. The question arises for each dimension d=0,1,2,.... We find p(d)
positive for d=0,1, and 2 and not positive for d=4. We conjecture that p(3) is
also positive. Similar questions may be phrased for (manifold, submanifold)
pairs and manifolds with other additional structure. The results in dimension 4
imply that unitary TQFTs cannot distinguish homotopy equivalent simply
connected 4-manifolds, nor can they distinguish smoothly s-cobordant
4-manifolds. This may illuminate the difficulties that have been met by several
authors in their attempts to formulate unitary TQFTs for d=3+1. There is a
further physical implication of this paper. Whereas 3-dimensional Chern-Simons
theory appears to be well-encoded within 2-dimensional quantum physics, eg in
the fractional quantum Hall effect, Donaldson-Seiberg-Witten theory cannot be
captured by a 3-dimensional quantum system. The positivity of the physical
Hilbert spaces means they cannot see null vectors of the universal pairing;
such vectors must map to zero.
