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  "body": "\n## Statement\n\nA path-connected space $X$ is *aspherical* when its [[homotopy groups|P976WYEFMB]] $\\pi_k(X)$ vanish for all $k \\ge 2$; equivalently, its [[universal cover|Y1YPE6PM52]] is contractible.\nLet $M$ and $N$ be closed connected aspherical [[topological manifolds|QQQ9B1DK4P]].\nThen every [[homotopy equivalence|XZG6MTSWCN]] $f \\colon M \\to N$ is homotopic to a [[homeomorphism|0QT00PB5G2]].\nIn particular, two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic.\n\n## Remarks\n\nThe conjecture is open in general.\nIt holds in dimensions at most $2$ by the classification of surfaces, and in dimension $3$ as a consequence of the geometrization theorem.\nFarrell and Jones proved it for closed Riemannian manifolds of nonpositive sectional curvature of dimension at least $5$.\nBartels and Lück proved it in dimensions at least $5$ when the fundamental group is hyperbolic or CAT(0).\nIn terms of the [[surgery exact sequence|Q7C480HQ9J]], it says that the structure set of a closed aspherical manifold of dimension at least $5$ has one element.\nThe analogue for diffeomorphisms is false: for $n \\ge 5$ there are smooth manifolds homeomorphic to the $n$-torus but not diffeomorphic to it.\n\n## Explanations\n\n### L1 human\n\nAn aspherical manifold is one whose shape is completely recorded by its loops: all higher-dimensional holes are absent.\nThe torus and closed hyperbolic manifolds are examples.\nThe conjecture says that for such manifolds the loops determine everything: if two of them have the same homotopy type, they are the same manifold up to homeomorphism.\n",
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