Rational generalized intersection homology theories

Given a spectrum $E$, we investigate the theory that associates to a stratified pseudomanifold the tensor product of its Goresky-MacPherson intersection homology with the rationalized coefficients of $E$. The viewpoint adopted in this paper is to express this theory as the homotopy groups of a spectrum associated to the pseudomanifold and $E$. The relation is given by an Atiyah-Hirzebruch formula. Properties such as topological invariance, generalized Poincaré duality, behavior under small resolution, products, cohomology operations, and the Künneth spectral sequence are then discussed from that viewpoint. Moreover, we consider self-dual generalized (co)homology theories on spaces that need not satisfy the Witt condition. Local calculations and a sample calculation of the rational intersection $ku$-theory of a certain singular Calabi-Yau 3-fold are carried out. We employ the framework of $S$-algebras and modules over Eilenberg-MacLane spectra due to Elmendorf, Kriz, Mandell and May.

Keywords

intersection homology, generalized homology theories, spectra, stable homotopy theory, self-dual sheaves, stratified spaces, singularities

2010 Mathematics Subject Classification

55N20, 55N33, 55P42

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Published 1 January 2010