TOPOLOGICAL INVARIANTS OF STRATIFIED SPACES. (1 of 2)

          Intersection homology theory provides a way to obtain generalized Poincaré duality,
          as well as a signature and characteristic classes, for singular spaces. For this to
          work, one has had to assume however that the space satisfies the so-called Witt
          condition. We extend this approach to constructing invariants to spaces more general
          than Witt spaces.

          We present an algebraic framework for extending generalized Poincaré duality and
          intersection homology to singular spaces X not necessarily Witt. The initial step in
          this program is to define the category SD(X) of complexes of sheaves suitable for
          studying intersection homology type invariants on non-Witt spaces. The objects in
          this category can be shown to be the closest possible self-dual "approximation'' to
          intersection chain sheaves. It is therefore desirable to understand the structure of
          such self-dual sheaves and to isolate the minimal data necessary to construct them.
          As the main tool in this analysis we introduce the notion of a Lagrangian structure.

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