Articles of spin geometry

Quaternion–Spinor relationship?

I’ve known for some time about the rotation group action of the (‘pure’) quaternions on $ \mathbf{R}^3 $ by conjugation. I’ve recently encountered spinors and notice similarities in their definitions (for example, the use of half-angles for rotations). Is the relationship that this suggested in my mind a real one, and if so what is […]

What does it mean that quaternions/ spinors are negated under a full rotation?

As I understand it, quaternions are a type of object called a spinor. Spinors are objects that are negated under a full rotation and only return to their original state under two full rotations. But what does that mean in the case of quaternions? A quaternion is a number $p=a+bi+cj+dk$ where $i^2=j^2=k^2=-1=ijk$. So it’s just […]

The last accidental spin groups

For dimensions $n\le 6$ there are accidental isomorphisms of spin groups with other Lie groups: $\DeclareMathOperator{Spin}{\mathrm{Spin}}$ $$\begin{array}{|l|l|} \hline \Spin(1) & \mathrm{O}(1) \\ \hline \Spin(2) & \mathrm{SO}(2) \\ \hline \Spin(3) & \mathrm{Sp}(1) \\ \hline \Spin(4) & \mathrm{Sp}(1)\times\mathrm{Sp}(1) \\ \hline \Spin(5) & \mathrm{Sp}(2) \\ \hline \Spin(6) & \mathrm{SU}(4) \\ \hline \end{array} $$ The definition of $\Spin(n)$ is […]

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the above is true? Note. More precisely, from Wikipedia: André Haefliger found necessary and sufficient conditions for the existence of a spin structure on […]