Computing twisted signatures and \(L\)-classes of stratified spaces
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 \(X\) [\textit{M. Goresky} and \textit{R. MacPherson}, Topology 19, 135--165 (1980; Zbl 0448.55004)]. The first formula above is combined with the Cappell-Shaneson signature formula [\textit{S. Cappell} and \textit{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 \(f\).
scientific article; zbMATH DE number 1982585
scientific article