Essential dimension lowering mappings having dense deficiency set

Two classes of surjective maps $f:{S^m} \to {S^n}$ that are one-to-one over the image of a dense set are constructed. We show that for $m,n \geq 3$ there is a monotone surjection $f:{S^m} \to {S^n}$ that is one-to-one over the image of a dense set; and for $3 \leq n \leq m \leq 2n - 3$, each element of ${\pi _m}({S^n})$ can be represented as a monotone surjection $f:{S^m} \to {S^n}$ that is one-to-one over the image of a dense set.