Intersection spaces, spatial homology truncation, and string theory.

*(English)*Zbl 1219.55001
Lecture Notes in Mathematics 1997. Dordrecht: Springer (ISBN 978-3-642-12588-1/pbk; 978-3-642-12589-8/ebook). xvi, 217 p. (2010).

Manifolds satisfy Poincaré duality, but pseudomanifolds need not. The intersection homology theory of M. Goresky and 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 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 [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 complementary perversities. The construction of intersection spaces is given for pseudomanifolds with isolated singularities and for some other pseudomanifolds with only two strata – in particular, those with arbitrary bottom stratum but trivial link bundle. As suggested by the author, it is likely that the theory will be found to be more widely applicable.

In general, \({H}_\ast(I^{\overline p}X; \mathbb{Q})\) is not isomorphic to \(IH_\ast^{\overline p}(X;\mathbb{Q})\), with a certain singular Calabi-Yau quintic providing an explicit counterexample. However, the two theories are closely related. For example, for singular Calabi-Yau \(3\)-folds, \((IH_\ast(-),H_\ast(I-))\) is a mirror pair in the sense of mirror symmetry in algebraic geometry. The author develops this connection in a separate chapter, which includes an application to type II string theory related to massless \(D\)-branes.

One advantage of the spatial approach is that classical functors of algebraic topology defined on spaces can, of course, be defined on intersection spaces, whereas the functors might not factor through chain complexes (including the intersection chain complex). For example, generalized homology theories such as \(K\)-theory, \(L\)-theory, stable homotopy groups, etc. may be applied to intersection spaces to obtain intersection versions of classical theories. Another advantage is the existence of products and cohomology operations on the ordinary cochain complex of the space \(I^{\overline p}X\), which do not exist for the intersection cohomology chain complex.

The construction of intersection spaces is based on the homotopy-theoretic technique of spatial homology truncation, developed in a lengthy chapter and of independent interest. Whereas spatial homology truncation has been studied on the object level (it is the Eckmann-Hilton dual of the Postnikov decomposition), Banagl concentrates on aspects of functorality.

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 [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 complementary perversities. The construction of intersection spaces is given for pseudomanifolds with isolated singularities and for some other pseudomanifolds with only two strata – in particular, those with arbitrary bottom stratum but trivial link bundle. As suggested by the author, it is likely that the theory will be found to be more widely applicable.

In general, \({H}_\ast(I^{\overline p}X; \mathbb{Q})\) is not isomorphic to \(IH_\ast^{\overline p}(X;\mathbb{Q})\), with a certain singular Calabi-Yau quintic providing an explicit counterexample. However, the two theories are closely related. For example, for singular Calabi-Yau \(3\)-folds, \((IH_\ast(-),H_\ast(I-))\) is a mirror pair in the sense of mirror symmetry in algebraic geometry. The author develops this connection in a separate chapter, which includes an application to type II string theory related to massless \(D\)-branes.

One advantage of the spatial approach is that classical functors of algebraic topology defined on spaces can, of course, be defined on intersection spaces, whereas the functors might not factor through chain complexes (including the intersection chain complex). For example, generalized homology theories such as \(K\)-theory, \(L\)-theory, stable homotopy groups, etc. may be applied to intersection spaces to obtain intersection versions of classical theories. Another advantage is the existence of products and cohomology operations on the ordinary cochain complex of the space \(I^{\overline p}X\), which do not exist for the intersection cohomology chain complex.

The construction of intersection spaces is based on the homotopy-theoretic technique of spatial homology truncation, developed in a lengthy chapter and of independent interest. Whereas spatial homology truncation has been studied on the object level (it is the Eckmann-Hilton dual of the Postnikov decomposition), Banagl concentrates on aspects of functorality.

Reviewer: Bruce Hughes (Nashville)

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

55N33 | Intersection homology and cohomology in algebraic topology |

57P10 | Poincaré duality spaces |

14J17 | Singularities of surfaces or higher-dimensional varieties |

55P30 | Eckmann-Hilton duality |

55S36 | Extension and compression of mappings in algebraic topology |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14J33 | Mirror symmetry (algebro-geometric aspects) |