Homology stability for orthogonal groups over algebraically closed fields. (English) Zbl 1133.20037
Let \(k\) be an algebraically closed field, or more generally an infinite Pythagorean field, of characteristic not two. One considers the integral homology of the orthogonal group \(\text{O}(n,k)\) associated with the quadratic form \(q_n(x)=\sum_{i=1}^nx_i^2\).
The main result is that the natural map \(H_i(\text{O}(n,k),\mathbb{Z})\to H^i(\text{O}(n+1,k),\mathbb{Z})\) is surjective for \(n=i\) and bijective for \(n\geq i+1\). There are several related theorems. For the groups \(\text{SO}(n,k)\) one must invert two and replace \(n\geq i+1\) with \(n\geq 2i+1\). If the field is quadratically closed the obstruction to homology stability for the \(\text{SO}(n,k)\), with 2 inverted, is given by a Milnor \(K\)-group.
To obtain sharp stability results the author studies homology of a so-called Steinberg module defined by means of a chain complex spanned by configurations of points and lines, in imitation of a Tits building. Things get complicated because of the existence of degenerate subspaces, such as isotropic subspaces. This makes that one cannot easily imitate the work done by Sah for the case of the real field. Several variant complexes need to be considered and compared.
The main result is that the natural map \(H_i(\text{O}(n,k),\mathbb{Z})\to H^i(\text{O}(n+1,k),\mathbb{Z})\) is surjective for \(n=i\) and bijective for \(n\geq i+1\). There are several related theorems. For the groups \(\text{SO}(n,k)\) one must invert two and replace \(n\geq i+1\) with \(n\geq 2i+1\). If the field is quadratically closed the obstruction to homology stability for the \(\text{SO}(n,k)\), with 2 inverted, is given by a Milnor \(K\)-group.
To obtain sharp stability results the author studies homology of a so-called Steinberg module defined by means of a chain complex spanned by configurations of points and lines, in imitation of a Tits building. Things get complicated because of the existence of degenerate subspaces, such as isotropic subspaces. This makes that one cannot easily imitate the work done by Sah for the case of the real field. Several variant complexes need to be considered and compared.
Reviewer: Wilberd van der Kallen (Utrecht)
MSC:
| 20G10 | Cohomology theory for linear algebraic groups |
| 19D45 | Higher symbols, Milnor \(K\)-theory |
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
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