If $M$ is an oriented $d$-manifold, let $D(M)$ denote the top exterior power of $H^*(M,\mathbf{C})$. Then $D(M_1 \amalg M_2) = D(M_1) \otimes D(M_2)$. Is there a good recipe for a map $D(M) \to D(N)$ induced by a cobordism from $M$ to $N$?

In some dimensions, there is a natural identification $D(M) = \mathbf{C}$ and you could take every cobordism to the identity map. But such a TQFT would not be completely trivial when $d = 4$, I think. For example it would distinguish $S^1 \times \mathbf{C}P^2$ from the mapping torus of complex conjugation.

If there's no problem defining it for $(d+1)$-manifolds, can it be "extended down" any distance, i.e. associate something to a $(d-1)$-manifold?