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Foliated Stratified Spaces and a De Rham Complex Describing Intersection Space Cohomology
Algebraic Topology · Submitted 23 Feb 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincar\'e 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincar\'e duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. | background 0.98 | 0.17 | 0.09 | asserted 1.00 |
| s1 | The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. | background 0.84 | 0.29 | 0.08 | asserted 1.00 |
| s2 | Mirror symmetry tends to interchange IH and HI. | background 0.81 | 0.07 | 0.03 | hedged possibility 0.91 |
| s3 | The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. | background 0.75 | 0.12 | 0.03 | asserted 0.76 |
| s4 | 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. | contribution 0.95 | 0.94 | 0.69 | asserted 1.00 |
| s5 | We prove that the wedge product of forms introduces a perversity-internal cup product on HI, for every perversity. | result 0.93 | 0.95 | 0.39 | asserted 1.00 |
| s6 | Flat link bundles arise for example in foliated stratified spaces and in reductive Borel-Serre compactifications of locally symmetric spaces. | background 0.90 | 0.15 | 0.03 | asserted 0.82 |
| s7 | A precise topological definition of the notion of a stratified foliation is given. | contribution 0.98 | 0.89 | 0.06 | asserted 1.00 |
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Original abstract
The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincar\'e 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.
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3 arXiv papers cite this in 12 sentences
Papers citing it · 9 sentences without a confident label
Uses its method or tool 2 papers
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The proof of Theorem [ref] relies on the complex of multiplicatively structured forms constructed in [ref] , and the fact that fiberwise truncation in both directions, yielding a subcomplex in both cases (and not a quotient complex in one case), can be performed on the multiplicatively structured forms.
Isometric Group Actions and the Cohomology of Flat Fiber Bundles · Introduction · 0.95
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In this subsection, we present the de Rham complex defined in [ref] , which computes the reduced singular cohomology of intersection spaces.
Hodge Theory for Intersection Space Cohomology · Extended Perversities and the de Rham Complex for $HI$ · 0.87
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