Bloch-ogus Properties for Topological Cycle Theory

In this paper, we re-formulate “morphic cohomology” as introduced by the author and H.B. Lawson [F-L1] in such a way that it and “Lawson homology” satisfy the list of basic properties codified by S. Bloch and A. Ogus [B-O]. This reformulation enables us to clarify and unify our previous definitions and provides this topological cycle theory with foundational properties which have proved useful for other cohomology theories. One formal consequence of these Bloch-Ogus properties is the existence of a local-to-global spectral sequence which should prove valuable for computations (as shown in Corollary 7.2). The basic result of this paper is that topological cycle cohomology theory (which agrees with morphic cohomology for smooth varieties) in conjunction with topological cycle homology theory (which is shown to always agree with Lawson homology) do indeed satisfy the Bloch-Ogus properties for a “Poincaré duality theory with supports” on complex quasi-projective varieties. We view the challenge of verification of the Bloch-Ogus properties as worthy for several reasons. First, the properties require certain definitions and constructions whose development add substance to morphic cohomology/Lawson homology. For example, considerable effort is required to extend earlier definitions to a cohomology theory defined and contravariantly functorial on all quasi-projective varieties. As another example, we formulate a cap product pairing which leads to a natural extension of earlier duality theorems of the author and Lawson [F-L2], [F2], [FL-4]. Second, the properties constrain the formulation of our theory, thereby giving us a good basis for choosing the definitions we propose. Third, the fact that these properties can be verified tells us that this theory, although originating as it does in differential geometry and topology, behaves very much as other theories familiar to algebraic geometers and thus might be more readily applicable to geometric problems. Finally, the constructions we present are closely related to those involved in formulating a suitable motivic cohomology theory as in [F-V] (for example, we use V. Voevodsky’s qfh-topology to extend definitions from normal varieties to all varieties), so that our topological point of view may serve as an accessible entry into that more algebraic theory. Over the past decade, Lawson homology and morphic cohomology have been reformulated several times, with each reformulation providing either a simplification of definitions and/or an extension of the class of varieties for which the theory is