Relationship between the Hopf Fibration and Spinors on $S^2$

The unique spin structre for $TS^2$ is given by the Hopf fibration. We can trivialize the Hopf fibration over open sets $U_1 = S^2 \setminus \{N\}, U_2 = S^2 \setminus \{S\}$ where $N$ and $S$ are the north and south pole respectively.

When I compute the transition function $T_{12}$ on $U_1 \cap U_2$ I get that $T_{12}$ does not depend on the height of the point on the sphere, and can be considered as the identity map $T_{12}: S^1 \rightarrow S^1.$

To get the transition functions for the bundle of spinors on $S^2,$ you compose the transition function for the Hopf bundle with the Spin representation of $Spin(2) = S^1.$ The Spin rep in this case is (I think) just the identification of $S^1$ with $U(1)$ which is a degree one map (at least this is what it comes out to be in Friedrich's book "Dirac Operators in Riemannian Geometry.") Thus it seems like the transition function for the bundle of spinors should be the degree one map $S^1 \rightarrow S^1,$ but according to many sources it should be the degree -1 map so that it can be identified with the canonical line bundle over $CP^1.$

Is the Spin representation of $S^1$ instead homotopic to a degree -1 map? If so, in what way is this canonical? Why should it be that way?

• Anyone have any ideas? – mck Dec 11 '12 at 15:24