Banagl, Markus; Cappell, Sylvain E.; Shaneson, Julius L. Computing twisted signatures and \(L\)-classes of stratified spaces. (English) Zbl 1034.32021 Math. Ann. 326, No. 3, 589-623 (2003). The main result are Atiyah-type characteristic class formulas for both the twisted signature \(\sigma(X;\mathcal S)\) and the twisted \(L\)-class \(L(X;\mathcal S)\) of a stratified space \(X\). They apply to a closed oriented Whitney stratified normal Witt space \(X\) of even dimension (for example, a compact normal complex algebraic variety). The twisting is encoded by a local coefficient system \(\mathcal S\) on the nonsingular set of \(X\) with a nondegenerate bilinear pairing. A condition (which is automatically satisfied if \(X\) is supernormal) of strong transversality of \(\mathcal S\) to the singular set of \(X\) is shown to imply that \(\mathcal S\) has a \(K\)-theory signature \([\mathcal S]_K\) in the \(K\)-theory of \(X\). The main formulas established are \[ \sigma(X;\mathcal S) = \varepsilon_\ast (\widetilde{\text{ch}}([\mathcal S]_K)\cap L(X)) \] and \[ L(X;\mathcal S) = \widetilde{\text{ch}}([\mathcal S]_K)\cap L(X) \] where \(\widetilde{\text{ch}}\) is a modified Chern character and \(L(X)\) is the Goresky-MacPherson \(L\)-class of \(X\) [M. Goresky and R. MacPherson, Topology 19, 135–165 (1980; Zbl 0448.55004)]. The first formula above is combined with the Cappell-Shaneson signature formula [S. Cappell and J. Shaneson, J. Am. Math. Soc. 4, 521–551 (1991; Zbl 0746.32016)] to give an expression for the signature of the source of a stratified map \(Y\to X\) (i.e., a map that is a fiber bundle projection over each stratum of \(X\)) between oriented compact Whitney stratified spaces with only even-codimensional strata, \(X\) normal and \(\dim (Y) - \dim (X)\) even, in terms of the \(L\)-classes of the closed strata of \(X\) and the Chern classes associated to the local coefficient systems induced on the strata of \(X\) by \(f\). Reviewer: Bruce Hughes (Nashville) Cited in 2 ReviewsCited in 12 Documents MSC: 32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants 32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) 55N25 Homology with local coefficients, equivariant cohomology 55N33 Intersection homology and cohomology in algebraic topology 55R55 Fiberings with singularities in algebraic topology 57N80 Stratifications in topological manifolds 57R20 Characteristic classes and numbers in differential topology 57R45 Singularities of differentiable mappings in differential topology 58A35 Stratified sets 58K10 Monodromy on manifolds Keywords:Whitney stratification; Witt space; signature; \(L\)-class; characteristic class; stratified map; stratified pseudomanifold; intersection homology Citations:Zbl 0448.55004; Zbl 0746.32016 PDFBibTeX XMLCite \textit{M. Banagl} et al., Math. Ann. 326, No. 3, 589--623 (2003; Zbl 1034.32021) Full Text: DOI