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Let $F =\mathbb{R}$ or $F=\mathbb{C}$. There is a close connection between the algebraic K-groups $K_i(F)$ and the topological K-groups $K^{-i}_F(P)$, where $P$ denotes the one-point space. I'm trying to learn this stuff at the moment so I hope someone can fill in the details here (post is community wiki), but I believe the statement is that if you take 2-completions the groups will be isomorphic.