We determine a nice simple formula for the largest Euclidean space for which there is an orientable n-manifold with a nonimmersion detected by Stiefel-Whitney classes. For Spin manifolds, we prove the analogue of the upper bound and establish the complete answer for $n\le 23$ and $32\le n\le 33$. Results similar to many of these were obtained some 50 years ago, but in a much less tractable form. The sharp results for Spin manifolds require detailed calculations of ko-homology groups of mod-2 Eilenberg MacLane spaces.

中文翻译：

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可定向流形和自旋流形的 Stiefel-Whitney 类和浸入式

我们为最大的欧几里得空间确定了一个很好的简单公式，其中有一个可定向的n流形，并且由 Stiefel-Whitney 类检测到非浸入。对于自旋流形，我们证明了上界的类似物并建立了完整的答案$n\le 23$ 和 $32\le n\le 33$. 大约 50 年前获得了与其中许多类似的结果，但形式要少得多。自旋流形的清晰结果需要对 mod-2 Eilenberg MacLane 空间的ko 同调群进行详细计算。