Abstract: It is known  that injective polynomial maps Cⁿ -> Cⁿ are surjective. In fact a theorem of Ax (1968) states that this is also true for injective regular maps X -> X, where X is an algebraic (possibly singular) variety over algebraically closed field of char =0. The same problem over the field of reals is more difficult. Bialynicki-Birula and Rosenlicht (1962) have proved that injective polynomial maps Rⁿ -> Rⁿ are surjective. But the general case of algebraic injective maps X-> X , where  X is real algebraic (possibly singular), was solved only recently by the author. The method is essentially topological (Borel-Moore homology), based on a fine decomposition of real algebraic sets into so called arc-symmetric components.