Blowup equivalence of smooth closed manifolds

J. Nash in [9] proved that every smooth closed manifold is diffeomorphic to a component of a real algebraic variety and concluded his paper with a question whether every smooth closed connected manifold is diffeomorphic to a rational real algebraic non-singular variety. The topological Nash conjecture formulated in [2, 4] arose from this question of Nash. The main result of this paper is the proof of the topological Nash conjecture (stated in Theorem 1.1).