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On topological invariants of stratified maps with non-Witt target. (English) Zbl 1094.57025

Let \(X\) be a closed, oriented, Whitney stratified pseudomanifold, then given a self dual complex of sheaves \(S^{\bullet}\) on \(X\), has homology \(L\)-classes defined by the Thom-Pontrjagin construction. If \(X\) is a Witt space then the middle-perversity intersection chain sheaf is self dual. If X is not Witt then there is a category of self dual sheaves \(SD(X)\) which can be understood in terms of Lagrangian structures along the strata of odd dimension. If \(SD(X) \neq \emptyset\) then one has a self dual sheaf and thus can again define classes \(L_{i}(X)\) that are independent of the choice of sheaf from \(SD(X)\). The main result of this paper is to show that for spaces with only odd dimensional strata any self dual sheaf is cobordant to an intersection sheaf associated to the top stratum. This is in contrast to the case of spaces with even codimensional strata where these lower dimensional strata do contribute [S. E. Cappell and L. J. Shaneson, J. Am. Math. Soc. 4, No. 3, 521–551 (1991; Zbl 0746.32016)]. The results are applied to exhibit a singular space which admits a PL resolution in the sense of M. Kato [Topology 12, 355–372 (1973; Zbl 0276.55014)], but has no resolution by a stratified map.

MSC:

57R20 Characteristic classes and numbers in differential topology
55N33 Intersection homology and cohomology in algebraic topology
57N80 Stratifications in topological manifolds
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
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