Spin^h and further generalisations of spin
Abstract
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 compact orientable manifold of dimension 7 or 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.
Keywords
Cite
@article{arxiv.2008.04934,
title = {Spin^h and further generalisations of spin},
author = {Michael Albanese and Aleksandar Milivojevic},
journal= {arXiv preprint arXiv:2008.04934},
year = {2023}
}
Comments
Incorporated content of corrigendum (https://doi.org/10.1016/j.geomphys.2022.104709), in particular Theorem 3.10. The question of whether every non-compact orientable manifold of dimension 6 or 7 is spin^h is equivalent to asking whether W_5 vanishes for such manifolds, which remains open