The classical example of a spinor is to consider a path around the center of a mobius strip, and attach to each point in the path a perpendicular vector that locally points "up". This carries the spinor property that passing the vector around in a full circle once, flips it, and twice restores it.
See this image here as an example:
Now I was trying to generalize the concept of a spinor by trying to find surfaces that do different activities to vectors than just flip them (When twisted around) and realized this is not so simple.
The mobius strip fundamental polygon sort of reveals why it behaves as a spinor, the identification of edges on the fundamental polygon involves a "flip" function $f: [0,1] \rightarrow [0,1]$. Which has the property $f(f(x)) = x$.
In fact there are only TWO homotopic classes of continuous functions from the line segment to the line segment. The identity and the flip (see the image below):
And thus if you build a surface by repeatedly gluing boundaries of some fundamental polygon you can only ever create a surface whose internal vector fields exhibits spinor like behavior
This suggests that if instead of looking at Surfaces (so called $2$-manifolds) if we instead look at $3$-manifolds, we might be able to create something interesting.
Consider a fundamental "triangular prism" like the one below:
We can glue the two triangles together in essentially 6 different ways (each way corresponding to a different symmetry of the triangle), the obvious way is to just identify them with each other with no transformation (The so called identity fusion). And I conjecture that if we glue them together with a reflection transformation on one of the triangles then we can get something akin to a spinor. But if we consider the rotation symmetries of a triangle, we should be able to create a $3$-manifold such that if you have loop going through the center of this manifold back to itself, and we attach a pair of vectors to each point that "locally" point up and to the left (similar to how the mobius strip centerline had a vector that always pointed up). THEN, if we loop this pair of vectors all the way around it will appear to have undergone a 1/3 twist. Creating a 3-spinor like object.
Does this work in practice? And if so does it have a name? If not: How can spinors be generalized to induce arbitrary group actions?