In perfect elastostatics, the unknown is the displacement $x\mapsto y$, where $x\in\Omega\subset{\mathbb R}^3$ is the reference configuration, and $y\in{\mathbb R}^3$. It obeys to an 2nd-order PDEs. When we rewrite the PDE as a 1st-order system, the unknown becomes the deformation gradient $F(x):=\nabla y$.

A fundamental constraint is that the matter cannot interpenetrate. In particular, one must have $\det F>0$ everywhere. This is highly non-trivial, in the sense that the set $P$ of $3\times3$ real matrices with positive determinant has a non-trivial homotopy type. Thanks to the polar factorization, $P$ is homeomorphic to the product of ${\bf SPD}_3$ (a convex cone, something trivial) with ${\bf SO}_3$ whose fundamental group is ${\mathbb Z}_2$.

This raises the following questions. Given a domain $\Omega\subset{\mathbb R}^3$, how many connected components are there in $C(\partial\Omega;P)$ ? How can we determine if an $f\in C(\partial\Omega;P)$ can be lifted as an $F\in C(\Omega;P)$ ? How can we determine if an $f\in C(\partial\Omega;P)$ can be lifted as an $F\in C(\Omega;P)$ such that $F$ is homotopic to the identity ?