The ACCT conjecture on the manifold structure of Ricci limit spaces

Let $X$ be a noncollapsed Ricci limit space of dimension $n$, with singular set ${{\cal S}}$. For each $k\in\{0,\ldots,n-1\}$, let ${{\cal S}}_k$ be the subset of ${{\cal S}}$ consisting of points at which no tangent cone can be written as a product of ${{\mathbb R}}^{k+1}$ with another space. The ACCT conjecture asserts that the ACCT conjecture. The interior of

$$X\setminus {{\cal S}}_{n-4}$$

is homeomorphic to a topological manifold. Cheeger–Colding theory gives that the Hausdorff dimension of ${{\cal S}}_k$ is at most $k$, and in the noncollapsed case ${{\cal S}}_{n-2}={{\cal S}}$. The conjecture seeks a topological-manifold structure away from the lower-dimensional singular stratum and remains unresolved in general.