Universal manifold pairings and positivity

Gluing two manifolds M-1 and M-2 with a common boundary S yields a closed manifold M. Extending to formal linear combinations x = Sigma 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] not equal 0 whenever x not equal 0. 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, e. g. 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.