We compute the Chow–Witt rings of the classifying spaces for the symplectic and special linear groups. In the structural description we give, contributions from real and complex realization are clearly visible. In particular, the computation of cohomology with ${\mathbf{I}}^j$‐coefficients is done closely along the lines of Brown's computation of integral cohomology for special orthogonal groups. The computations for the symplectic groups show that Chow–Witt groups are a symplectically oriented ring cohomology theory. Using our computations for special linear groups, we also discuss the question when an oriented vector bundle of odd‐rank splits off a trivial summand.

中文翻译：

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辛和特殊线性群分类空间的Chow–Witt环

我们为辛和特殊线性群计算分类空间的Chow–Witt环。在我们给出的结构描述中，来自真实和复杂实现的贡献清晰可见。特别是与${\mathbf{一世}}^{Ĵ}$系数是按照布朗对特殊正交群的积分同调计算得出的。辛群的计算表明，Chow–Witt群是一种面向环的同调理论。使用我们对特殊线性组的计算，我们还讨论了当奇数秩的定向向量束平凡求和时的问题。