We introduce a method that associates to a singular space a
CW complex whose ordinary rational homology satisfies
Poincaré duality across complementary perversities as in intersection
homology. The method is based on a homotopy theoretic
process of spatial homology truncation, whose functoriality properties
are investigated in detail. The resulting homology theory is not
isomorphic to intersection homology and addresses certain questions
in type II string theory related to massless D-branes.
The two theories satisfy an interchange of third and second plus fourth
Betti number for mirror symmetric conifold transitions.
Further applications of the new theory to K-theory and symmetric L-theory