Scissors congruences and the bar and cobar constructions.
In this article, the cohomology of Dehn complexes, as considered by \textit{A. Goncharov} [J. Am. Math. Soc. 12, No. 2, 569--618 (1999; Zbl 0919.11080)], is related to the homology of isometry groups, considered as discrete groups, for the three classical geometries and in odd dimensions. The machinery developed is based on the interpretation of a Hopf algebra of spherical polytopes in terms of shuffle algebras. Certain bar and cobar constructions play a fundamental role. By extending a result of J. Dupont for \(m=2\), the author proves, among other things, that the cohomology of the Dehn complex \({\mathcal D}(m)\), in degrees \(m\) and \(m-1\), is given by \[ \begin{aligned} H^m({\mathcal D}(m))&\cong H_m(O(2m,{\mathbb R}),{\mathbb Q}^t);\\ H^{m-1}({\mathcal D}(m)) &\cong H_{m+1}(O(2m,{\mathbb R}),{\mathbb Q}^t), \end{aligned} \] where \(\,H_{*}(O(n,{\mathbb R}),{\mathbb Q}^t)\,\) denotes the homology of the orthogonal group \(O(n,{\mathbb R})\), considered as discrete group, acting on \({\mathbb Q}\) by the determinant. This yields the known cohomology of the Dehn complex of \(S^3\). A related result is that the natural map \[ \bigotimes\limits^m H_{1}(O(2,{\mathbb R}),{\mathbb Q}^t) \rightarrow H_{m}(O(2m,{\mathbb R}),{\mathbb Q}^t) \] is surjective. These two results show up some analogy with the appearance of Milnor K-Theory in motivic cohomology. The author provides also a description of \(H^1({\mathcal D}(3))\) as a quotient of the homology group \(\,H_5(O(6,{\mathbb R}),{\mathbb Q}^t)\). This reduces Hilbert's third problem for \(S^5\), up to torsion, to the study of the homology of \(O(6,{\mathbb R})\), considered as discrete group. At the end, counterparts of these results in hyperbolic and euclidean geometry are presented.
scientific article; zbMATH DE number 1930180
scientific article