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The method of intersection spaces associates cell-complexes depending on aperversity to certain types of stratified pseudomanifolds in such a way thatPoincar\'e duality holds between the ordinary rational cohomology groups of thecell-complexes associated to complementary perversities. The cohomology ofthese intersection spaces defines a cohomology theory HI for singular spaces,which is not isomorphic to intersection cohomology IH. Mirror symmetry tends tointerchange IH and HI. The theory IH can be tied to type IIA string theory,while HI can be tied to IIB theory. For pseudomanifolds with stratificationdepth 1 and flat link bundles, the present paper provides a de Rham-theoreticdescription of the theory HI by a complex of global smooth differential formson the top stratum. We prove that the wedge product of forms introduces aperversity-internal cup product on HI, for every perversity. Flat link bundlesarise for example in foliated stratified spaces and in reductive Borel-Serrecompactifications of locally symmetric spaces. A precise topological definitionof the notion of a stratified foliation is given.

Foliated Stratified Spaces and a De Rham Complex Describing Intersection Space Cohomology