This is a very naive question. I have the impression that the area of "Spin geometry" is not an active research field. Sure Spin geometry is used in many different branches of mathematics and physics as a tool, but I don't see papers published on the development of Spin geometry by itself. Is this true? Loosely speaking, by "Spin geometry" I mean here the area in mathematics that would have the "Spin geometry" book of Lawson and Michelsohn as the basic reference. 


Are there any open questions in Spin geometry?


Thanks.



Open questions in "Spin geometry"
As @314159 has explained, Spin geometry on "spin manifolds" is a very active field of research, but the subject goes way beyond the domain spin manifolds. The key point to notice is that every pseudo-Riemannian spin manifold $(M,g)$ admits a bundle of irreducible Clifford modules over the bundle of Clifford algebras $Cl(M,g)$, which is usually called the "spinor bundle", but remarkably enough, the opposite is not true. In other words, there exist manifolds which admit "spinor bundles" but do not admit any spin structure. This is already clear just by looking at the literature on $Spin^{c}$ manifolds, but it turns out the possibilities go beyond $Spin^{c}$ manifolds as well. In fact, so you can see a very recent paper on the subject, the following paper by C. Lazaroiu and Carlos Shahbazi:


<https://arxiv.org/abs/1606.07894>


classifies all the spinorial structures associated to a bundle of irreducible real Clifford modules, and gives the topological obstructions for the existence of such bundles on any pseudo-Riemannian manifold. You can check that these obstructions are in general different from that required by a spin structure, and correspond to what they call a "Lipschitz structure". They even have some application of his formalism to physics. This proves that "spin geometry" is not always associated to a spin structure and indeed it is a very rich and subtle construction. I think spin geometry in this more general setting is very little explored, and I believe is going to attract a lot of attention in the future.


