COMPUTATION OF TWISTED CHARACTERISTIC CLASSES OF SINGULAR SPACES.

          In ongoing joint work with Sylvain Cappell and Julius Shaneson, we establish characteristic
          class formulae for both the twisted signature and the twisted L-classes of stratified
          spaces. The local system is assumed to extend into the singularities. For the twisted signature
          of manifolds, this formula reduces to a prototypical formula due to Atiyah and W. Meyer.
          We obtain applications to stratified maps by combining our formula with the Cappell-Shaneson
          signature formula. The current focus of our efforts is the geometrically dual situation
          of (nonlocally flat PL) embeddings of stratified spaces.

          Banagl showed that the above formulae hold even when the stratified space does not
          satisfy the Witt condition. This required the use of a homology theory called
          signature homology, constructed by Augusto Minatta and Matthias Kreck.

         If the local systems do not extend into the singularities, then we have examples that show
         that the above formulae become false. Building on recent results of Banagl, we obtain formulae
         for nonlocally flat embeddings that involve rho-invariants, even when the local systems do not
         extend.         

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