K-Theory of Intersection Spaces

The construction of an intersection space assigns to certain pseudomanifolds a topological space, called intersection space. This intersection space depends on a perversity and the reduced homology with rational coefficients of the intersection space satisfies Poincaré duality across complementary perversities. Therefore, by modifications on a spatial level, this construction restores Poincaré duality for stratified pseudomanifolds. We extend Poincaré duality for certain intersection spaces as shown by M. Banagl to a broader class of intersection spaces coming from two-strata pseudomanifolds whose link bundles allow a fiberwise truncation. Further properties of this class of intersection spaces are discussed, including the existence of cap products and a calculation of the signature. J.F. Adams shows that Poincaré duality for manifolds can be generalized to any homology theory given by a CW-spectrum. We combine these two approaches and show Poincaré duality in complex K-theory for intersection spaces coming from a suitable class of pseudomanifolds, including the class of two strata pseudomanifold mentioned above. Finally, for pseudomanifolds with only isolated singularities, an approach is given, where the spatial homology truncation is performed with respect to any homology theory given by a connective ring spectrum. The objects constructed are not CW-complexes, but CW-spectra. Their rational homology equals intersection homology