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Techniques, computations, and conjectures for semi-topological K-theory

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Abstract.

We establish the existence of an “Atiyah-Hirzebruch-like” spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that relates the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) homology of complex varieties introduced by H. Gillet and C. Soulé – to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulate integral conjectures relating semi-topological K-theory to topological K-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod-n algebraic K-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above.

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Correspondence to Christian Haesemeyer.

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Mathematics Subject Classification (2000): 19E20, 19E15, 14F43

The first author was partially supported by the NSF and the NSA

The second author was supported by the Helen M. Galvin Fellowship of Northwestern University

The third author was partially supported by the NSF and the NSA

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Friedlander, E., Haesemeyer, C. & Walker, M. Techniques, computations, and conjectures for semi-topological K-theory. Math. Ann. 330, 759–807 (2004). https://doi.org/10.1007/s00208-004-0569-3

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