The signature package on Witt spaces (Q2903104)

On closed oriented manifolds, there is a very successfull interplay between (higher) signatures and (higher) index theory of the signature operator, based on the good analytic properties of the signature operator. The authors call this the ``signature package''.NEWLINENEWLINEThe paper at hand extends this, in particular all the analytic aspects, to a ``Witt space'' \(X\), i.e., a stratified pseudo-manifold. Roughly speaking, this means that it is the union of strata and the complement of one stratum in the next one is a manifold (with regularity conditions on how they are glued together). The appropriate cohomology theory in this context is ``intersection cohomology''. The Witt condition now requires that each even dimensional link has vanishing intersection homology in middle degree. Important examples are arbitrary complex algebraic varieties.NEWLINENEWLINEThere are canonical classes of Riemannian metrics on the regular part of the Witt space \(X\), with which one can define the signature operator \(\partial_X\), and also the de Rham operators. Indeed, one can also twist these with (flat) Hilbert \(A\)-module bundles for a \(C^*\)-algebra \(A\), like the Mishchenko bundle (where \(A=C^*\Gamma\) for the fundamental group \(\Gamma\)).NEWLINENEWLINEThe main results of the paper are the following.NEWLINENEWLINEThe signature operator on a Witt space is essentially self adjoint, and the \(A\)-twisted one regular in the sense of Hilbert \(A\)-module operators.NEWLINENEWLINEThe spectral properties of a (twisted) signature operator are those of an elliptic operator on a compact manifold, in particular the index (for the \(A\)-twisted version as an element in \(K_*(A)\)) is defined.NEWLINENEWLINESimilarly, the analytic properties allow to define from the signature operator an analytic K-homology class, the signature class. The K-theoretic index is obtained by the Kasparov product of the signature class with the K-theory class of the Hilbert \(A\)-module bundle.NEWLINENEWLINEUsing KK-theoretic methods, the authors show Witt-bordism invariance of the \(C^*\)-index of the twisted signature operator.NEWLINENEWLINEThe authors then prove with analytic methods that the K-theoretic index is a stratified homotopy invariant. The proof is based on the strategy of Hilsum and Skandalis.NEWLINENEWLINEA corollary of all of this is then that higher topological signatures (defined as in the case of closed manifolds from the total L-class) are stratified homotopy invariants, provided the Baum-Connes assembly map for the fundamental group \(\Gamma\) is rationally injective. In particular, the topological Novikov conjecture for Witt spaces follows from the analytic Novikov conjecture for the fundamental group.NEWLINENEWLINETo establish the basic properties of the signature operators, the authors develop microlocal techniques, which have the advantage to generalize to the \(A\)-twisted version. \textit{J. Cheeger} [Proc. Symp. Pure Math., Vol. 36, 91--146 (1980; Zbl 0461.58002); Proc. Natl. Acad. Sci. USA 76, 2103--2106 (1979; Zbl 0411.58003)] established the corresponding results for the untwisted operator using heat kernel methods, and with a somewhat sketchy presentation.NEWLINENEWLINEThe microlocal analysis is very involved, the basic idea is to work inductively on the strata.