For a Keller map, invertible, injective and proper are the same

This is the closed-universe theorem: a Keller map that loses nothing to infinity is invertible.

• (iii) ⇒ (ii). A proper local biholomorphism of ℂⁿ is a covering map. Since ℂⁿ is simply connected, the covering has one sheet (the Hadamard–Caccioppoli global inverse function theorem).
• (ii) ⇒ (i). An injective polynomial self-map of ℂⁿ is bijective (Ax 1968; Grothendieck, EGA IV) and has a polynomial inverse (Cynk–Rusek 1991; Rudin 1995).^[1]
• (ii) ⇒ (iv). An injective étale map has one-point generic fibers, so it is birational.
• (iv) ⇒ (i). This is Keller's birational theorem, which turns a birational Keller map into an automorphism.

So the Jacobian conjecture says every Keller map of ℂⁿ is proper, and every counterexample in dimension three and up is a Keller map that is not proper. Where properness fails is the non-properness set.

Neither hypothesis can be dropped casually. Pinchuk's real maps are not proper. The entire map $(e^x,ye^{-x})$ has Jacobian 1 and is periodic in $x$ with period $2\pi i$, so it is not injective.

1. ↑ S. Cynk, K. Rusek, Injective endomorphisms of algebraic and analytic sets, Ann. Polon. Math. 56 (1991) 29–35; Ax–Grothendieck theorem, Wikipedia.