Homology, Homotopy and Applications

Rational generalized intersection homology theories

Markus Banagl

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Abstract

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.

Article information

Source
Homology Homotopy Appl., Volume 12, Number 1 (2010), 157-185.

Dates
First available in Project Euclid: 28 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.hha/1296223826

Mathematical Reviews number (MathSciNet)
MR2607414

Zentralblatt MATH identifier
1200.55009

Subjects
Primary: 55N33: Intersection homology and cohomology 55N20: Generalized (extraordinary) homology and cohomology theories 55P42: Stable homotopy theory, spectra

Keywords
Intersection homology generalized homology theories spectra stable homotopy theory self-dual sheaves stratified spaces singularities

Citation

Banagl, Markus. Rational generalized intersection homology theories. Homology Homotopy Appl. 12 (2010), no. 1, 157--185. https://projecteuclid.org/euclid.hha/1296223826


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