Subject: K-theory Date: Thu, 6 Sep 2001 15:42:10 -0500 From: Eric Friedlander Because of some mis-references and confusion concerning the query about algebraic and topological vector bundles on complex varieties, I provide the following brief discussion. In addition to the papers referenced earlier by others, interested mathematicians can learn more by consulting the following recent papers by Mark Walker and myself available on the K-theory preprint server (and on our homepages). E. Friedlander and M. Walker, Function spaces and continuous algebraic pairings for varieties, Compositio Math. {\bf 125} (2001), 69-110. E. Friedlander and M. Walker, Semi-topological $K$-theory using function complexes. To appear in Topology (2001), 54 pages. E. Friedlander and M. Walker, Comparing $K$ theories for complex varieties. To appear in the American Journal (2001), 32 pages. E. Friedlander and M. Walker, Semi-topological $K$-theory of real varieties. Presumably, this has appeared in the Proceedings of the Bombay Colloquium, 2000. (83 pages). E. Friedlander and M. Walker, Some remarks concerning mod-$n$ $K$-theory. To appear in Inventiones (2001), 11 pages. First of all, I shall restrict attention to complex varieties although analogous statements (involving Atiyah's Real K-theory) are valid for real varieties. Second, I restrict attention to quasi-projective varieties (i.e., those that are locally closed in some complex projective space with respect to the Zariski topology). Third, we consistently work with spectra (sometimes using operads and sometimes $\Gamma$-spaces), but I shall be more informal and discuss only associated homotopy groups. Finally, Mark and I have in draft a paper which addresses certain issues concerning rational K-groups and which is needed to justify a few of the statements I make below. To answer the initial question, the map from algebraic K-theory (in degree 0) to topological K-theory can have uncountable kernel -- even for projective smooth curves. For any integer $n$, we can find projective smooth varieties such that the cokernel of this map has rank at least $n$. This cokernel arises from the phenomenon that not all (singular) cohomology of a variety need be algebraic (i.e., arise as fundamental classes of subvarieties). Mark Walker and I introduce "semi-topological K-theory" which interpolates between algebraic and topological K-theory $$K_*(X) \to K_*(X \times \Delta_{top}^{\bullet}) \to K_{top}^{-*}(X^{an})$$ where $X^{an}$ is the analytic space underlying the complex variety $X$. Among our various theorems, we show that the map from algebraic K-theory mod-$n$ to semi-topological K-theory mod-$n$ is always an isomorphism in all degrees and that the map from semi-topological K-theory to topological K-theory becomes an isomorphism for smooth varieties once we invert the Bott element in semi-topological K-theory. Rationally, we understand this map of K-theories by passing to "cohomology": to Suslin-Voevodsky motivic cohomology for algebraic K-theory, to Friedlander-Lawson morphic cohomology for semi-topological K-theory, and to singular cohomology for topological K-theory. In degree 0, it is quite classical that the Chern character determines a rational isomorphism for each context (algebraic/semi-topological/topological). Indeed, one has rational isomorphisms in all degrees, not simply degree 0. Finally, a word of advertisement. Semi-topological K-theory can be viewed as the result of imposing algebraic equivalence (a familiar relation for algebraic cycles) on algebraic K-theory. One could say that semi-topological K-theory retains the geometrically relevant information of algebraic K-theory: it agrees with algebraic K-theory mod-$n$ and it provides a filtration which is conjecturally equivalent to the Hodge filtration on the cohomology of smooth varieties. On the other hand, it discards the "irrelevant": the algebraic K-theory of the complex numbers $Spec C$ contains an uncountable rational vector space in every positive degree which is not present in the semi-topological K-theory of $Spec C$. best wishes, Eric Eric M. Friedlander