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Recently, the norm-residue homomorphism has been the subject of intense discussions in the K-theoretic community following the proof of the Bloch-Kato conjecture by Voevodsky, Suslin and Rost (see Rost’s lecture at this year’s Arbeitstagung.) The goal of this post is to explain the norm-residue homomorphism in a down to earth ring-theoretic language.

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Recently I was trying to understand what’s behind the mysterious condition

$\sum_i [z_i]\wedge[1-z_i] = 0\in \Lambda^2 \mathbb C^\times$

for elements $z_i\in \mathbb C$ ($z_i\neq 0, 1$) to define an element $\sum_i[z_i]$ in the Bloch group. It appears that the condition naturally appears if one studies hyperbolic $3$-manifolds.