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 (ν + 1)th Betti number of their union is strictly greater than the sum of their (ν + 1)th Betti numbers, where ν = dim M + dim K - dim N. This result is applied to the study of coincidence sets and fixed point sets.
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