TAILIEUCHUNG - Đề tài " The L-class of non-Witt spaces "

Characteristic classes for oriented pseudomanifolds can be defined using appropriate self-dual complexes of sheaves. On non-Witt spaces, self-dual complexes compatible to intersection homology are determined by choices of Lagrangian structures at the strata of odd codimension. We prove that the associated signature and L-classes are independent of the choice of Lagrangian structures, so that singular spaces with odd codimensional strata, such as . certain compactifications of locally symmetric spaces, have well-defined L-classes, provided Lagrangian structures exist. We illustrate the general results with the example of the reductive Borel-Serre compactification of a Hilbert modular surface. . | Annals of Mathematics The L-class of non-Witt spaces By Markus Banagl Annals of Mathematics 163 2006 743 766 The L-class of non-Witt spaces By Markus Banagl Abstract Characteristic classes for oriented pseudomanifolds can be defined using appropriate self-dual complexes of sheaves. On non-Witt spaces self-dual complexes compatible to intersection homology are determined by choices of Lagrangian structures at the strata of odd codimension. We prove that the associated signature and L-classes are independent of the choice of Lagrangian structures so that singular spaces with odd codimensional strata such as . certain compactifications of locally symmetric spaces have well-defined L-classes provided Lagrangian structures exist. We illustrate the general results with the example of the reductive Borel-Serre compactification of a Hilbert modular surface. Contents 1. Introduction 2. The Postnikov system of Lagrangian structures 3. The bordism group Q D 4. The signature of non-Witt spaces 5. The L-class of non-Witt spaces 6. An example References 1. Introduction Finding natural settings for defining characteristic classes has been and continues to be an important theme in geometry. The notion of multiplicative sequences allowed Hirzebruch Hir56 the definition of L-classes in rational cohomology as certain polynomials in the Pontrjagin classes leading to his beautiful formula stating equality of the signature and L-genus of a smooth oriented manifold. Using this result together with the principle of representing The author was in part supported by NSF Grant DMS-0072550. 744 MARKUS BANAGL cohomology classes by transverse maps to spheres Thom Tho58 constructed L-classes for triangulated manifolds which are piecewise linear invariants. To define L-classes for singular spaces various approaches have been successful in various settings. In GM80 Goresky and MacPherson introduce intersection homology theory as a method to recover generalized Poincare duality for stratified .

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