Stable splitting of mapping spaces via nonabelian Poincar\'e duality

We use nonabelian Poincar\'e duality to recover the stable splitting of compactly supported mapping spaces, $\rm{Map_c}$$(M,\Sigma^nX)$, where $M$ is a parallelizable $n$-manifold. Our method for deriving this splitting is new, and naturally extends to give a more general stable splitting of the space of compactly supported sections of a certain bundle on $M$ with fibers $\Sigma^nX$, twisted by the tangent bundle of $M$. This generalization incorporates possible $O(n)$-actions on $X$ as well as accommodating non-parallelizable manifolds.