Orbit spaces of unimodular rows over smooth real affine algebras
The main result is as follows: Let \(X=\text{Spec}(R)\) be a smooth real affine variety of even dimension \(d\geq2\), whose real points \(X(\mathbb R)\) constitute an orientable manifold. Then the set of isomorphism classes of (oriented) stably free modules of rank \(d\) is a free abelian group of rank equal to the number of compact connected components of \(X(\mathbb R)\). In contrast, if \(d\geq3\) is odd, then the set of isomorphism classes of (oriented) stably free modules of rank \(d\) is a \(\mathbb Z/2\mathbb Z\)-vector space (possibly trivial). That the set of isomorphism classes of (oriented) stably free modules of rank \(d\) is an abelian group is known from \textit{Wilberd van der Kallen} [J. Algebra 82, 363--397 (1983; Zbl 0518.20035)], but these groups \(Um_{d+1}(R)/\mathrm{SL}_{d+1}(R)\) have been hard to compute. Thus the authors have to combine many ingredients to arrive at the main result. They make a good effort to explain these ingedients to a general reader. One compares \(Um_{d+1}(R)/\mathrm{SL}_{d+1}(R)\) with a group \(Um_{d+1}(\mathbb R(X))/E_{d+1}(\mathbb R(X))\), where \(\mathbb R(X)\) is a convenient localization of \(R\). One also compares it with the Euler class groups \(E^d(R)\) and \(E^d(\mathbb R(X))\). In fact the authors employ homomorphisms in both directions between \(Um_{d+1}(R)/E_{d+1}(R)\) and \(E^d(R)\). The one from \(E^d(R)\) to \(Um_{d+1}(R)/E_{d+1}(R)\) is more recent. It deals with naive homotopy classes -- also known as homotopy orbits -- in a quadric over \(R\) with equation \(z^2+\sum_ix_iy_i=1\). This relies on \textit{A. Stavrova} [J. K-Theory 13, 199--248 (2014; Zbl 1314.19002)]. Higher Mennicke symbols and \(K\)-cohomology are also discussed.
scientific article; zbMATH DE number 6860750
scientific article