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I'm trying to read some differential geometry these days and I just encountered orientable manifold.

Quote: "If $M$ is nonorientable, $M$ has a two-sheeted orientable covering manifold $\tilde{M}$. If $M$ is simply connected, then $M$ is orientable."

I heard my instructor talking during the class couple of days before where he mentioned the spin/spinor lives in a spaces of two copy.

Question 1: what does orientable manifold has to do with spinor?

Question 2: what kind of space does spinor lives in? as it seemed to imply that every spinor got a "hold" or singular value inside of them.

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  • $\begingroup$ why here and not math.SE? $\endgroup$ Feb 3 '20 at 1:52
  • $\begingroup$ @AccidentalFourierTransform I don't think math people do spinor or electron like that in physics. $\endgroup$ Feb 3 '20 at 1:56
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    $\begingroup$ When you ask “what kind of space”, what kind of explanation do you want? Spinor space is a linear vector space that acts as a representation space in the sense of group representation theory. It is not a geometric space like spacetime. $\endgroup$
    – G. Smith
    Feb 3 '20 at 5:07
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    $\begingroup$ What do you mean by “hold”? $\endgroup$
    – G. Smith
    Feb 3 '20 at 5:09
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Question 1: what does orientable manifold has to do with spinor?

Spinor fields cannot be consistently defined on an arbitrary manifold. As the Encyclopedia of Mathematics explains, “Necessary and sufficient conditions for the existence of a spinor structure on $M$ consist of the orientability of $M$ and the vanishing of the Stiefel–Whitney class $W_2(M)$.

(To understand what a Stiefel-Whitney class is, take a course in algebraic topology or ask on Math SE.)

Question 2: what kind of space does spinor lives in?

Spinor space is a complex linear vector space that acts as a representation space for the Poincaré group in the sense of group representation theory. It is not a geometric space like spacetime on which fields are defined; it is a space in which certain fermion fields have their values. Instead of having four real dimensions like spacetime, spinor space has two or four complex dimensions.

A Poincaré transformation of spacetime coordinates makes the spinor field at each point change its “length” and “direction” in the spinor space. Suppose transformation $A$ in spacetime followed by transformation $B$ is equivalent to a single transformation $C$. Then the transformation $A’$ in spinor space corresponding to $A$, followed by the transformation $B’$ corresponding to $B$, must be equivalent to the single transformation $C’$ corresponding to $C$. In other words, the transformations of field values in spinor space are homomorphic to the transformations of coordinates in spacetime. Both “represent” the same abstract group composition law, on two completely different spaces!

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