The question of which manifolds are spin or spin${}^c$ has a simple and complete answer. In this paper we address the same question for spin${}^h$ manifolds, which are less studied but have appeared in geometry and physics in recent decades. We determine that the first obstruction to being spin${}^h$ is the fifth integral Stiefel–Whitney class $W_5$. Moreover, we show that every orientable manifold of dimension 7 and lower is spin${}^h$, and that there are orientable manifolds which are not spin${}^h$ in all higher dimensions. We are then led to consider an infinite sequence of generalised spin structures. In doing so, we show that there is no integer $k$ such that every manifold embeds in a spin manifold with codimension $k$.

哪个歧管旋转或旋转的问题${}^C$有一个简单而完整的答案。在本文中，我们针对自旋解决了相同的问题${}^H$流形，研究较少，但近几十年来出现在几何学和物理学中。我们确定旋转的第一个障碍${}^H$ 是斯蒂芬·惠特尼课程的第五部分 ${\mathrm{w\; ^}}_5$。而且，我们表明尺寸为7或更低的每个可定向流形都是自旋的${}^H$，并且存在不旋转的可定向歧管${}^H$在所有更高的维度上。然后，我们被引导去考虑一个无限的广义自旋结构序列。这样做，我们表明不存在整数$ķ$ 这样每个流形都以同维数嵌入自旋流形 $ķ$。