### Techniques, Computations, and Conjectures for Semi-Topological K-theory, by Eric M. Friedlander, Christian Haesemeyer, and Mark E. Walker

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 connects 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)
cohomology of complex varieties introduced by H. Gillet and C. Soule --- 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.

Eric M. Friedlander <eric@math.northwestern.edu>

Christian Haesemeyer <chh@math.northwestern.edu>

Mark E. Walker <mwalker@math.unl.edu>