Stable homotopy methods for solving differential relations

The homotopy principle (h-principle) is a general observation that differential relations in differential geometry can often be solved by means of homotopy theory. For example, in the case of immersions this implies the Smale paradox asserting that the standard embedding of the sphere $S^2$ into $R^3$ can be turned inside out through a family of immersions. According to a general theorem of Gromov, the h-principle holds true in a very general setting. Often solutions of a differential relation R form a monoid under operation defined by taking the disjoint union. In fact, often the moduli space of solutions of R is a Segal $\Gamma$-space. In particular, its group completion is an infinite loop space. I will determine the homotopy type of such infinite loop spaces. For example, I will show that the group completion of the moduli space of coverings is the space $\Omega^{\infty}S^{\infty}$, which recovers the Barratt-Priddy-Quillen theorem. I will also show explicit computations in several non-obvious cases.