Regularity at infinity of real mappings and a Morse-Sard theorem

We prove a new Morse-Sard type theorem for the asymptotic critical values of semi-algebraic mappings and a new fibration theorem at infinity for $C^2$ mappings. We show the equivalence of three different types of regularity conditions which have been used in the literature in order to control the asymptotic behaviour of mappings. The central role of our picture is played by the $t$-regularity and its bridge toward the $\rho$-regularity which implies topological triviality at infinity.