Transfer Maps and Orientation Classes of Singular Spaces

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The confluence of topological methods such as L-theory, intersection homology and ad-theories, complex algebraic methods coming from the theory of mixed Hodge modules, and global real analytic methods involving iterated edge Riemannian metrics and K-homology, has recently lead to the construction of a dense portfolio of orientation and characteristic classes for singular spaces that often generalize classical invariants of manifold theory. Understanding the behavior of these classes under various topological and algebraic Gysin restrictions, bundle transfer maps and pullback under smooth algebraic morphisms begins to emerge as a key to computability and applicability. The project thus aims for discerning and establishing the basic transformational laws governing orientation classes of singular spaces under transfers on several relevant homology theories such as bordism, K- and L-homology. The results are expected to impact existing conjectures concerning the relation of the invariants to each other. The laws found will then provide the foundation of new computational methods for concrete types of singular spaces, particularly projective varieties beyond rational homology manifolds, e.g. for Schubert varieties and other varieties arising in mathematical Physics.