Baum–Douglas elliptic boundary condition conjecture

Let $M$ be a compact oriented smooth manifold with smooth boundary $\partial M$, let $E_0$ and $E_1$ be smooth complex Hermitian vector bundles over $M$, and let $D:C^{\infty}(E_0)\to C^{\infty}(E_1)$ be a first-order elliptic differential operator. Let $[D]\in K_0(M,\partial M)$ be its relative $K$-cycle, and let

$$\partial:K_0(M,\partial M)\longrightarrow K_1(\partial M)$$

be the boundary map. Baum–Douglas elliptic boundary condition conjecture. There exist a vector bundle $E_2$ over $\partial M$ and a zeroth-order pseudodifferential operator $B:C^{\infty}(\partial M,E_0)\to C^{\infty}(\partial M,E_2)$ such that

$$\begin{array}{l} \left(\begin{array}{c} D\\ B\circ\gamma\end{array}\right):\ H^1(M,E_0)\longrightarrow\begin{array}{c} H^0(M,E_1)\\ \oplus\\ H^{1/2}(\partial M,E_2)\end{array} \end{array}$$

is Fredholm if and only if $\partial[D]=0$ in $K_1(\partial M)$, where $\gamma:H^1(M,E_0)\to H^{1/2}(\partial M,E_0)$ is the trace map. This conjecture identifies the vanishing of the boundary of the relative $K$-cycle as the only obstruction to the existence of an elliptic boundary condition; the source states that the “if” direction is proved when $\dim(M)\not=4,5,6,7$ and the “only if” direction in arbitrary dimension.