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Singular spaces and generalized Poincaré complexes. (English) Zbl 1215.55003

The ordinary homology of a pseudomanifold \(X\) does not in general satisfy Poincaré duality; the intersection homology groups \(IH_\ast^{\overline p}(X)\) of M. Goresky and R. MacPherson [Topology 19, 135–165 (1980; Zbl 0448.55004) and Invent. Math. 72, 77–129 (1983; Zbl 0529.55007)] were designed to correct the lack of duality. Whereas the intersection homology groups are the homology groups of a subcomplex of the ordinary chain complex of \(X\), the current paper introduces a new approach at the space level. The goal is to associate to a closed oriented pseudomanifold \(X\) spaces \(I^{\overline p}X\), called the intersection spaces of \(X\), that are generalized rational Poincaré complexes in the sense that the ordinary rational homology and cohomology groups satisfy \(\widetilde{H}^i(I^{\overline p}X; \mathbb{Q} ) \cong \widetilde{H}_{n-i}(I^{\overline q}X; \mathbb{Q})\), where \(X\) is an \(n\)-dimensional oriented closed pseudomanifold and \(\overline{p}\) and \(\overline{q}\) are complementary perversities. This association is outlined in the very special cases of isolated singularities and for some other pseudomanifolds with only two strata. A fuller treatment appears in the monograph [M. Banagl, Intersection spaces, spatial homology truncation, and string theory. Lecture Notes in Mathematics 1997. Dordrecht: Springer. (2010; Zbl 1219.55001)], but even there the theory does not yet apply to all pseudomanifolds. One advantage to this novel spatial approach emphasized by the author is the existence of internal cup products. An application to type II string theory related to massless \(D\)-branes is mentioned.

MSC:

55N33 Intersection homology and cohomology in algebraic topology
57P10 Poincaré duality spaces
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
55P30 Eckmann-Hilton duality
55S36 Extension and compression of mappings in algebraic topology
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