Homology of orthogonal groups: A quadratic algebra. (English) Zbl 1047.18013

Cathelineau, Jean-Louis

\(K\)-Theory 30, No. 1, 13-35 (2003).

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.

Reviewer: Victor P. Snaith (Hants)

MSC:

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

Keywords:

homology of orthogonal groups with twisted coefficients; Steinberg modules; scissors congruence groups; bar and cobar constructions