Does there exist a prime 3-manifold such that its mapping class group has an abelian representation in which the 2$\pi$ rotation is represented by -1?

In detail:
Let $M$ be a closed orientable prime 3-manifold.
Let $D_F(M,p)$ be the group of diffeomorphisms of $M$ that fix a point $p$ of $M$ and a frame there. Define the mapping class group (MCG) of $M$ to be the zeroth homotopy group of $D_F(M,p)$. Then the $2\pi$ rotation is an element of MCG that is the equivalence class of the following diffeo, $R_{2\pi}$:
Consider a coordinate ball of radius 2, $B2$, centred on $p$. $R_{2\pi}$ fixes the ball of radius 1, $B_1$, centred on $p$ and everything outside the sphere of radius 2. In between $B_1$ and $B_2$ the $R_{2\pi}$ maps $(x,y,z)\rightarrow(x\cos\theta+y\sin\theta,y\cos\theta−x\sin\theta,z)$ where $\theta$ is a function of $r=\sqrt(x^2+y^2+z^2)$ which increases smoothly and monotonically from 0 to 2$\pi$ as $r$ increases from 1 to 2. The square of the 2$\pi$ rotation is the identity in MCG.
A manifold is *spinorial* if $\[R_{2\pi}\]$ is non-trivial in MCG.

**Background motivation:**

This question is interesting because of the possibility that fermions can be built on non-trivial spatial topology. $M$ is the manifold of a 3-D spatial hypersurface in spacetime. The fixed point is the point at infinity (where the metric is asymptotically flat) and fixing a frame there has the same effect on $\pi_0(D_F)$ as requiring some falloff conditions on the diffeomorphisms at infinity or requiring them to be the identity outside some ball. The configuration space of General Relativity in this asymptotically flat setting is (space of asymptotically flat metrics on $M$)/$D_F$ and its first homotopy group is isomorphic to (what I called above) MCG, see http://arxiv.org/abs/math-ph/0606066 (I know it is not the usual definition of MCG). The quantum state, on canonical quantisation of General Relativity, carries a unitary irreducible representation (UIR) of the MCG and different choices of UIR give different physics. Prime 3-manifolds are potentially candidates for elementary particles built from pure geometry: topological geons (this is speculative!). A prime 3-manifold can be the basis for a spinorial particle (i.e. spin 1/2, spin 3/2 ....) if $R_{2\pi}$ is nontrivial. Because particles must be able to be pair produced and annihilated, topology change must be allowed in the theory which means that the theory should be quantised in a sum-over-histories framework rather than a canonical quantisation framework. Within the sum-over-histories framework it is challenging to realise nonabelian reps of MCG. Abelian reps on the other hand are more easily accommodated by attaching phases to topologically distinct sectors of the path integral. Moreover certain rules that would result in a spin-statistics correlation for topological geons would also force the reps to be abelian, http://arxiv.org/abs/gr-qc/9609064 (hence the need for abelian reps). However, if there were no spinorial primes with abelian reps this would rule out spinorial geons and therefore fermions.