Intersection spaces, spatial homology truncation, and string theory
type: Item
instance of: scholarly article
title: Intersection spaces, spatial homology truncation, and string theory
zbMATH Open document ID: 1219.55001
DOI: 10.1007/978-3-642-12589-8
author: Markus Banagl
published in: Lecture Notes in Mathematics
publication date: 2010-05-27T00:00:00Z
review text: Manifolds satisfy Poincaré duality, but pseudomanifolds need not. The intersection homology theory of \textit{M. Goresky} and \textit{R. MacPherson} [Topology 19, 135--165 (1980; Zbl 0448.55004); Invent. Math. 72, 77--129 (1983; Zbl 0529.55007)] and the \(L^2\)-cohomology theory of \textit{J. Cheeger} [Proc. Natl. Acad. Sci. USA 76, 2103--2106 (1979; Zbl 0411.58003)] correct the lack of duality for some singular spaces. For a closed, oriented pseudomanifold \(X\), the intersection homology groups \(IH_\ast^{\overline p}(X;\mathbb{Q})\), which depend on a perversity parameter \(\overline p\), are the homology groups of a subcomplex of the ordinary chain complex of \(X\), and satisfy generalized Poincaré duality across complementary perversities. If \(X\) is additionally triangulated with a conical Riemannian metric on the top stratum, then the \(L^2\)-cohomology groups \(H^\ast_{(2)}(X)\) are the cohomology groups of the chain complex of differential \(L^2\)-forms on \(X\), and satisfy Poincaré duality. The linear dual of \(IH_{\ast}^{\overline{m}}(X;\mathbb{R})\) is known to be isomorphic to \(H^\ast_{(2)}(X)\), where \(\overline{m}\) is the middle perversity. Neither intersection homology nor \(L^2\)-cohomology theory provides the structure of a differential graded algebra on the chain complex because of problems with the cup and wedge products. The author provides a new approach at the space level, previously announced in [\textit{M. Banagl}, Electron. Res. Announc. Math. Sci. 16, 63--73 (2009; Zbl 1215.55003)]. For some \(X\), he constructs 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 complementa…
reviewed by: Bruce Hughes
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 81T30
Mathematics Subject Classification ID: 55N33
Mathematics Subject Classification ID: 57P10
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 55P30
Mathematics Subject Classification ID: 55S36
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 14J33
zbMATH DE Number: 5712714
zbMATH Keywords: intersection homology
zbMATH Keywords: intersection spaces
zbMATH Keywords: spatial homology truncation
zbMATH Keywords: string theory
zbMATH Keywords: Eckmann-Hilton duality
zbMATH Keywords: Moore space decomposition
zbMATH Keywords: Calabi-Yau manifolds
zbMATH Keywords: conifold
zbMATH Keywords: Poincaré duality
zbMATH Keywords: rational Poincaré complexes
zbMATH Keywords: mirror pair of homology theories
zbMATH Keywords: massless \(D\)-branes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-12589-8
OpenAlex ID: W1530982877
description: scientific article; zbMATH DE number 5712714