I am wondering if we can compute $\pi_0(Homeo(S^3))$ (i.e. the group of hoemomorphisms of the three-sphere mod isotopy) or if anyone has a reference where I could find such information.
2026-04-29 09:38:58.1777455538
Mapping Class Group of $S^3$
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Hatcher proved that the group $Homeo(S^3)$ is homotopy equivalent to the orthogonal group. Furthermore, in all dimensions Alexander's trick shows that two homeomorphisms of spheres are isotopic iff they have the same degree (1 or -1).