All compact manifolds are homeomorphic to totally algebraic real algebraic sets

The real algebraic set we obtain will in general be singular. The homeomorphism will be piecewise differentiable. Total algebraicity is a very useful concept since it eiliminates many obstructions to making a topological situation algebraic, c.f. [AK1], [AK2], [AK5]. At first glance our result does not seem to be very useful since current algebraic approximation lemmas require nonsingularity. The value of this paper is really in a more complex situation. In particular, our singular algebraic set has a resolution of singularities which is totally algebraic. Thus in practice one would apply the existing approximation theorems to the nonsingular resolved manifold and then blow down to the manifold in which you are really interested. For a relatively simple example, suppose that you have a smooth m a p f : N ~ M between smooth manifolds that you wish to make algebraic. Perhaps M is such that it has no totally algebraic model, thus standard techniques do not hold. By Theorem 8 below there is an algebraic multiblowup ~ : Z ~ Y and a homeomorphism h : M ~ Y with Z totally algebraic. Suppose that h f : N , Y approximately lifts to a map g : N * Z , so 7rg ~ hf. (Perhaps this was arranged after blowing up N, a common procedure in [AK2].) Then by standard algebraic approximation theorems, we may assume N is a nonsingular real algebraic set and g is an entire rational function. In particular, f : N -~ M is approximated by the algebraic situation ng : N ~ Y. In doing so, M has