On the Betti number of the union of two generic map images

Let f:M→N and g:K→N be generic differentiable maps of compact manifolds without boundary into a manifold such that their intersection satisfies a certain transversality condition. We show, under a certain cohomological condition, that if the images f(M) and g(K) intersect, then the (v+1)th Betti number of their union is strictly greater than the sum of their (v+1)th Betti numbers, where v=dim M+dim K−dim N. This result is applied to the study of coincidence sets and fixed point sets.