Abstract
The question of which manifolds are spin or spinc has a simple and complete answer. In this paper we address the same question for spinh manifolds, which are less studied but have appeared in geometry and physics in recent decades. We determine that the first obstruction to being spinh is the fifth integral Stiefel–Whitney class W5. Moreover, we show that every orientable manifold of dimension 7 and lower is spinh, and that there are orientable manifolds which are not spinh 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.
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