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Projective configurations, homology of orthogonal groups, and Milnor \(K\)-theory. (English) Zbl 1057.19002

Let \(k\) be an algebraically closed field of characteristic different from 2. In the paper the author gives a connection between homology of orthogonal groups (considered as discrete groups) and Milnor \(K\)-theory. This is done via A. Goncharov’s scissors Hopf algebra \(S_*(k)\) [J. Am. Math. Soc. 12, 569-618 (1999; Zbl 0919.11080)]. Let \(O(n,k)\) be an orthogonal group and \({\mathbb Q}^t\) denote the \(O(n,k)\)-module \({\mathbb Q}\otimes{\mathbb Z}^t\), where \({\mathbb Z}^t\) is \({\mathbb Z}\) endowed with the action of \(g\in O(n,k)\) given by multiplication by \(\det\,g\). The author studies the direct sum: \({\bigoplus}_{m\geq 0} H_m(O(2m,k),{\mathbb Q}^{t})\). This becomes a graded commutative algebra under the product given by composition of the natural maps: \(H_m(O(2m,k),{\mathbb Q}^{t})\otimes H_n(O(2n,k),{\mathbb Q}^{t})\rightarrow H_{m+n}(O(2m,k)\times O(2n,k),{\mathbb Q}^{t}\otimes {\mathbb Q}^{t})\rightarrow H_{m+n}(O(2(m+n),k),{\mathbb Q}^{t})\). Let \(K_*^Mk_{\mathbb Q}= K_{*}^M(k)\otimes {\mathbb Q}\), where \(K_*^M(k)\) is the Milnor \(K\)-theory of \(k\). One of the main theorems of the paper is the following theorem: The two graded \({\mathbb Q}\)-algebras \(K_{*}^M(k)_{\mathbb Q}\) and \({\bigoplus}_{m\geq 0} H_m(O(2m,k),{\mathbb Q}^{t})\) are canonically isomorphic.
Since \(K_{*}^M(k)\) has generators in degree one with relations in degree 2 the author proves the corresponding fact for the algebra \({\bigoplus}_{m\geq 0} H_m(O(2m,k),{\mathbb Q}^{t})\). In the proof the commutative Hopf algebra of projective configurations \(S_{*}(k)\), considered by Goncharov (loc.cit), is used. Let \({\mathcal G}(m,k)\) denote the reduced cobar complex for \(S_{m}(k)\). The author proves the following theorem: The two graded \({\mathbb Q}\)-algebras \[ {\bigoplus}_{m\geq 0} H_m({\mathcal G}(m,k)) \qquad\text{and} \qquad {\bigoplus}_{m\geq 0} H_m(O(2m,k),{\mathbb Q}^{t}) \] are canonically isomorphic.

MSC:

19D45 Higher symbols, Milnor \(K\)-theory
14L35 Classical groups (algebro-geometric aspects)
20G10 Cohomology theory for linear algebraic groups

Citations:

Zbl 0919.11080
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References:

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