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Oriented cohomology, Borel-Moore homology, and algebraic cobordism. (English) Zbl 1193.14027

The author first presents a generalization of the oriented cohomology theory of I. Panin [Homology Homotopy Appl. 11, No. 1, 349–405 (2009; Zbl 1169.14016)], where he generalizes Panin’s integration to integration with support. Paired with this notion of oriented cohomology theory, the authors introduces oriented Borel-Moore homology theory and put forward the axioms for an oriented duality theory. Such theories differ from the usual Bloch-Ogus twisted duality theories in that one only requires for line bundles \(L\) and \(M\) \[ c_1(L\otimes M)=F_A(c_1(L),c_1(M)) \] to be given by a formal group law.
Following M. Mocanasu, the authors shows that to each Panin’s theory, there is a unique extension to an oriented duality theory. At the end, the application to algebraic cobordism is presented. The author shows that there is a unique natural transformation from the “geometric” algebraic cobordism in M. Levine and F. Morel [Algebraic cobordism. Berlin: Springer (2007; Zbl 1188.14015)] to the Borel-Moore homology of any oriented duality pair.
In particular, such a natural transformation exists for the Borel-Moore theory \(\text{MGL}'\) associated to Voevodsky’s algebraic cobordism represented by the motivic Thom spectra. Then he conjectures this transformation on homology level is an isomorphism, and argues there are more techniques to establish this homology version. The details of that proof appeared in [M. Levine, J. Algebra 322, No. 9, 3291–3317 (2009; Zbl 1191.14023)].

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
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