Stable topological invariants for soft sphere representations in the tenfold way classes

Let $H_r$, for $r=1,\ldots,d+1$, be matrices in one of the ten random-matrix-theory universality classes that give a soft representation of the sphere $S^d$. In the stable limit, consider the topological invariants of these soft representations. Stable-invariant conjecture. If the $H_r$ belong to the GUE, GOE, or GSE classes, then for dimensions $2$ through $9$ the only topological invariants in the stable limit are those described in the tables for the GUE, GSE, and GOE classes, respectively. This conjecture predicts that the stable matrix invariants in these three classes agree with the corresponding free-fermion results. The paper explains that the conjecture is needed only in some applications to free-fermion systems and later gives a K-theoretic calculation of the invariants in all dimensions; the supplied text does not establish the conjecture.