Spiegel, Matthias Jonas K-theory of intersection spaces. (English) Zbl 1284.55011 Heidelberg: Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät (Diss.). v, 105 p. (2013). Summary: The construction of an intersection space [M. Banagl, Intersection spaces, spatial homology truncation, and string theory. Lecture Notes in Mathematics 1997. Dordrecht: Springer. xvi, 217 p. (2010; Zbl 1219.55001)] 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 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. In [J. F. Adams, Stable homotopy and generalised homology. Chicago Lectures in Mathematics. Chicago-London: The University of Chicago Press. X, 373 p. (1974; Zbl 0309.55016)] Poincaré duality for manifolds is 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. Cited in 3 Documents MSC: 55N33 Intersection homology and cohomology in algebraic topology 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory 57P99 Generalized manifolds Keywords:stratified pseudomanifold; intersection space; Poincaré duality; computed \(K\)-theory Citations:Zbl 1219.55001; Zbl 0309.55016 PDFBibTeX XMLCite \textit{M. J. Spiegel}, K-theory of intersection spaces. Heidelberg: Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät (Diss.) (2013; Zbl 1284.55011) Full Text: Link