A simple C$^*$-algebra which is not $K_1$-injective
We exhibit a simple unital infinite-dimensional C$^*$-algebra $A$ which is not $K_1$-injective, i.e., the canonical map $ U(A)/U_0(A)\longrightarrow K_1(A)$ is not injective. This phenomenon is well known in the non-simple case. Indeed, a natural example is the generator $u$ of $π_4(U(2))\cong\mathbb{Z}/2$, and we use exactly this example as the seed of our construction. The construction proceeds by developing a higher-rank matrix trapping technique in homogeneous algebras generalizing the machinery of Villadsen in higher stable rank simple C$^*$-algebras. A hypothetical homotopy from the homomorphic image of $u$ to the identity in a certain homogeneous algebra is shown, through matrix trapping, to yield a spin bordism from the class of $u$ to that of the identity matrix $\mathrm{1}_2 \in M_2$, and this, in turn, is obstructed by the fact that the bordism class of $u$ is not trivial. We then fit our new arguments into a generalized version of Villadsen's higher stable rank construction to arrive at a simple example.