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Homology of orthogonal groups: A quadratic algebra. (English) Zbl 1047.18013
Let \((E, {\mathbf q})\) denote a non-degenerate finite-dimensional quadratic space over an infinite field \(k\) of characteristic different from two. Let \(O(E, {\mathbf q})\) denote the corresponding orthogonal group and let \({\mathbb Q}^{t}\) denote the rational numbers on which the orthogonal group acts via the determinant homomorphism. An orientiation of \((E, {\mathbf q})\) is an orbit under the special orthogonal group of an orthonormal basis for the scalar extension to an algebraic closure of \(k\).
It is shown that \(H_{i}( (E, {\mathbf q}) ; {\mathbb Q}^{t})\) vanishes for all \(i\) if \(E\) is odd-dimensional and for \(i < \text{ dim}(E)/2\) otherwise. Next the author studies the first potentially non-zero homology group. Define a rational vector space \({\mathbf B}(k)\) to be the direct sum of the homology groups \(H_{\dim(E)/2}( (E, {\mathbf q}) ; {\mathbb Q}^{t})\) where \((E, {\mathbf q})\) runs through the isometry classes of even-dimensional forms.
The second main result is to show that \({\mathbf B}(k)\) is a connected, graded-commutative quadratic algebra where, in the terminology of rational homotopy theory, a \({\mathbb Q}\)-algebra is quadratic if it is generated by one-dimensional classes modulo relations of dimension two. The author uses a characterisation of quadratic algebras via Hochschild homology to prove this result.
In addition connections with scissors congruence cohomology groups and homology with Steinberg module coefficients are derived en route.

18G60 Other (co)homology theories (MSC2010)
20G10 Cohomology theory for linear algebraic groups
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