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A Nash cohomology class on a compact Nash manifold is a mod 2 cohomology class whose Poincaré dual homology class can be represented by a Nash subset. We find a canonical way to define Nash cohomology classes on an arbitrary compact smooth manifold M. Then the Nash cohomology ring of M is compared to the ring of algebraic cohomology classes on algebraic models of M. This is related to three conjectures concerning algebraic cohomology classes.
We study triviality of Nash families of proper Nash submersions or, in a more general
setting, the triviality of pairs of proper Nash submersions. We work with Nash manifolds
and mappings defined over an arbitrary real closed field . To substitute the
integration of vector fields, we study the fibers of such families on points of the real
spectrum and we construct models of proper Nash submersions over
smaller real closed fields. Finally we obtain results on finiteness of topological types
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