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Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete topology. Weibel, in his $K$-theory book, computes the torsion in its coefficient ring. I would like to know the torsion-free part in the homotopy groups, but can't find this anywhere. The best language for this might be in terms of motives (without factoring out $\mathbb{A}^1$), but I don't know where to find its homotopy groups computed in this language either. Anyone know this?

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  • $\begingroup$ When you say "minor modifications to deal with uncountable-dimensionality" are you thinking of completing at a prime, or something else? $\endgroup$ – Dan Ramras May 13 '14 at 15:04
  • $\begingroup$ There are probably no modifications necessary in this case - thanks. And I'm interested in the torsion-free part, so fine with taking coefficients in any characteristic-zero field. Edited question accordingly. $\endgroup$ – Dmitry Vaintrob May 14 '14 at 3:49
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I think the answer to this question is not known. All we can say about the K-theory of $\mathbb{C}$ concerns the torsion. The trouble starts with $K_1(\mathbb{C})\cong\mathbb{C}^\times$, which is pretty difficult to understand as an abelian group. There is a formula for $K_2$ of a field due to Matsumoto which is $K_2(F)\cong (F^\times\otimes F^\times)/(x\otimes(1-x)|x\in F^\times\setminus\{1\})$ which you probably found in Weibel's book already. For $K_3$ we still have a computation due to Suslin, which expresses $K_3(\mathbb{C})$ in terms of a Bloch group, this can be found in the book "Homology of linear groups" by K.Knudson but probably also in Weibel's K-book. This description relates $K_3(\mathbb{C})$ to scissors congruences in hyperbolic space and the three-sphere. However, although there is such a conceptual interpretation of $K_3$, it is still far from understood: it is (I believe) still an open question if the natural inclusion $K_3(\overline{\mathbb{Q}})\to K_3(\mathbb{C})$ is surjective, this is a rigidity question attributed to Sah in Dupont's book on scissors congruences. Note that $K_3(\overline{\mathbb{Q}})$ is "understood" in terms of Borel regulators. Finally, there is a (still conjectural) generalization of the Bloch-group description of $K_3$ to higher K-groups, due to Goncharov. You can find this for instance in Goncharov's article in the second part of Motives (Proceedings of Symposia in Pure Mathematics, Volume 55, eds Jannsen, Kleiman, Serre, 1994), or other articles of Goncharov.

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According to Corollary 22.4 here: http://www.math.uiuc.edu/~dan/Papers/KTheoryOfFields.pdf the K-theory of an algebraically closed field is divisible. According to the "structure theorem for divisible abelian groups", discussed here: http://homepages.math.uic.edu/~gconant/Math/Structure%20Theorem%20for%20Divisible%20Abelian%20Groups.pdf a divisible group is a direct sum of copies of $\mathbb{Q}$ and Prufer $p$-groups. (I think this theorem is discussed in L. Fuchs' book(s) on abelian group theory.) The remaining question, as far as the torsion-free part goes, seems to be the number of $\mathbb{Q}$ factors in each dimension (as a cardinal).

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