Corrigendum to “Spin and further generalisations of spin” [J. Geom. Phys. 164 (2021) 104174]

Spin^h Manifolds

“…See, for example, [9,21]. Recent math papers include [1,2,12,13,18]. Now in fact, it turned out that physicists had also found Spin h -manifolds, and were interested in them for physical reasons.…”

Generalised Spin$^r$ Structures on Homogeneous Spaces

“…Similarly, the Wu manifold W = SU(3)/ SO(3) is not Spin C , but it is Spin H [7]. Moreover, there are examples of manifolds which are not even Spin H , for instance W × W [2].…”

“…These structures are crucial for Seiberg-Witten theory, which has become an essential tool in the study of smooth 4-manifolds [23]. In 2021, Albanese and Milivojević [2,3] proved that every oriented Riemannian manifold of dimension ≤ 5 is Spin H . If, moreover, the manifold is closed, this is true for dimension ≤ 7.…”

“…Their associated spinor bundles have been studied in depth in [10,15,14]. Albanese and Milivojević [2] call these structures generalised Spin k structures. An oriented Riemannian manifold M is Spin r if, and only if, it embeds into a Spin manifold with codimension r. Hence, if r < s, every Spin r manifold is Spin s .…”

“…For each n ∈ N, we give simple examples of n-dimensional oriented Riemannian homogeneous Gspaces M for which the minimal r ∈ N such that M admits a G-invariant Spin r structure is exactly n. This provides a G-invariant analogue of the fact that, for each r ∈ N, there exists an oriented Riemannian manifold which does not admit a Spin r structure [2].…”