## Zagier’s conjecture and wedge complexes in algebraic $$K$$-theory.(English)Zbl 0868.19002

In “Polylogarithms, Dedekind zetafunctions, and the algebraic $$K$$-theory of fields” [in: Arithmetic algebraic geometry, Prog. Math. 89, 391-430 (1991; Zbl 0728.11062)], D. Zagier formulated a conjecture about the higher odd $$K$$-theory groups $$K_{2n-1}(F)$$ of a number field $$F$$ based on Bloch’s earlier work in the case $$n=2$$. Beilinson and Deligne reformulated the conjecture as follows:
Let $${\mathcal L}=F^*\otimes \mathbb{Q}$$ and let $$\{x\}_1= 1-x$$ for $$x\in F\setminus \{0,1\}$$. There should exist abelian groups $${\mathcal L}_n$$ for $$n\geq 1$$ generated by symbols $$\{x\}_n$$, $$x\in F\setminus \{0,1\}$$ together with a map $$d_n:{\mathcal L}\to\bigwedge^2 (\bigoplus^{n-1}_{l=1}{\mathcal L}_l)$$ given by $$\{x\}_n\mapsto x\wedge\{ x\}_{n-1}$$. There should exist an isomorphism $$\varphi_n:\ker d_n\to K^{(n)}_{2n-1}(F)$$ and the composite of $$\varphi_n$$ and the Borel regulator map should be given explicitly in terms of the $$n$$-th polylogarithm.
Using a formalism of multi-relative $$K$$-theory the author constructs for each $$n\geq 1$$ complexes $$\widetilde{\mathcal M}_{(n)}$$ of $$\mathbb{Q}$$-vector spaces, whose $$p$$-th cohomology $$H^p(\widetilde{\mathcal M}_{(n)})$$ maps to $$K^{(n)}_{2n-p}(F)$$. This leads to a proof of part of the Zagier conjecture for all $$n\geq 2$$, since for $$p=1$$ this map is an injection. Moreover, using work of Suslin, Goncharov and Zagier, the author shows that for $$n=2,3$$ surjectivity holds as well, i.e., the maps $$H^1(\widetilde{\mathcal M}_{(n)})\to K^{(n)}_{2n-1}(F)$$ are isomorphisms for $$n=2,3.$$ The same result holds for all $$n\geq 2$$ provided $$F=\mathbb{Q}(\zeta_N)$$ is a cyclotomic field.

### MSC:

 19F15 Symbols and arithmetic ($$K$$-theoretic aspects) 11R70 $$K$$-theory of global fields 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 11R42 Zeta functions and $$L$$-functions of number fields

Zbl 0728.11062
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### References:

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