Lefschetz coincidence theorem ($L(f,g)\ne 0⟹\exists m:f(m)=g(m)$)

A nonzero Lefschetz coincidence number forces a coincidence of the two maps. If their paired map misses the diagonal, the diagonal Thom class restricts to zero on its image and its pullback vanishes, making the coincidence number zero. The argument works for continuous maps and requires neither isolated coincidences nor transversality.