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Singular spaces, characteristic classes, and intersection homology. (English) Zbl 0759.55002

This paper obtains a calculation of the intersection homology \(L\)-class defined by Goresky and MacPherson. This is obtained in the context of PL stratified spaces \(X\) with all strata of even codimension and the stratification topologically locally trivial, though the PL category is used for technical convenience only as the proofs use perverse sheaves. The first half of the paper develops the background intersection homology theory needed in some detail. Care is required since the coefficients for homology need not to be taken in local systems over the strata.
An important feature is a detailed treatment of duality. In particular, since this involves varying the perversity, a ‘peripheral complex’ representing the difference of the two middle perversities comes to the fore, and the first major result states that this is cobordant to a direct sum of components supported on the strata, the sum being orthogonal relative to a natural self-pairing. This is a special case of a general result on perverse self-dual torsion sheaves.
The main theorem supposes \(X\) embedded with codimension 2 in a manifold, and obtains the difference of the \(L\)-class of \(X\) from that given by the formula for the case \(X\) nonsingular as a sum over the strata of \(L\)- classes with appropriate local pairings as coefficients. Under various mild conditions, the terms of the sum can be put into a computable form.
The result is novel and impressively general, and the authors apply it to signatures of nonlocally flat knots and to singular hypersurfaces in complex projective space. The proofs demand a considerable degree of expertise, and a judicious combination of known techniques.

MSC:

55N33 Intersection homology and cohomology in algebraic topology
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57R20 Characteristic classes and numbers in differential topology
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