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The h-cobordism theorem is a noted theorem in differential and PL topology. Ageneralization of the h-cobordism theorem for possibly non simply connectedmanifolds is the so called s-cobordism theorem. In this paper, we provesemialgebraic and Nash versions of these theorems. That is, starting withsemialgebraic or Nash cobordism data, we get a semialgebraic homeomorphism(respectively a Nash diffeomorphism). The main tools used are semialgebraictriangulation and Nash approximation. One aspect of the algebraic nature ofsemialgebraic or Nash objects is that one can measure their complexities. Weshow h and s-cobordism theorems with a uniform bound on the complexity of thesemialgebraic homeomorphism (or Nash diffeomorphism) obtained in terms of thecomplexity of the cobordism data. The uniform bound of semialgebraich-cobordism cannot be recursive, which gives another example of noneffectiveness in real algebraic geometry see [ABB]. Finally we deduce thevalidity of the semialgebraic and Nash versions of these theorems over any realclosed field.

h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash categories, Uniform bound and Effectiveeness