Journal of Differential Geometry

Foliated stratified spaces and a De Rham complex describing intersection space cohomology

Markus Banagl

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Abstract

The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincaré duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with stratification depth 1 and flat link bundles, the present paper provides a de Rham-theoretic description of the theory HI by a complex of global smooth differential forms on the top stratum. We prove that the wedge product of forms introduces a perversity-internal cup product on HI, for every perversity. Flat link bundles arise for example in foliated stratified spaces and in reductive Borel–Serre compactifications of locally symmetric spaces. A precise topological definition of the notion of a stratified foliation is given.

Article information

Source
J. Differential Geom. Volume 104, Number 1 (2016), 1-58.

Dates
First available in Project Euclid: 6 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1473186538

Digital Object Identifier
doi:10.4310/jdg/1473186538

Mathematical Reviews number (MathSciNet)
MR3544285

Zentralblatt MATH identifier
06645284

Citation

Banagl, Markus. Foliated stratified spaces and a De Rham complex describing intersection space cohomology. J. Differential Geom. 104 (2016), no. 1, 1--58. doi:10.4310/jdg/1473186538. https://projecteuclid.org/euclid.jdg/1473186538


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