The period isomorphism in tame geometry

We describe singular homology of a manifold $X$ via simplices $\sigma:\Delta_d\to X$ that satisfy Stokes' formula with respect to all differential forms. The notion is geared to the case of tame geometry (definable manifolds with respect to an o-minimal structure), where it gives a description of the period pairing with de Rham cohomology via definable $\sigma$'s.