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 in...
Let be a real-analytic submanifold and H(M) the algebra of real analytic functions on M. If K ⊂ M is a compact subset we consider ; is a multiplicative subset of . Let be the localization of H(M) with respect to . In this paper we prove, first, that is a regular ring (hence noetherian) and use this result in two situations: 1) For each open subset , we denote by O(Ω) the subalgebra of H(Ω) defined as follows: f ∈ O(Ω) if and only if for all x ∈ Ω, the germ of f at x, , is algebraic...