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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.

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