Actions of Manifold Homeomorphism Groups on Closed Three-Manifolds
Continuous actions of compact Lie groups on manifolds were extensively studied in the 1950s and 60s. A major milestone was Mostert’s 1957 classification of all such actions with a codimension-one orbit, which, as a corollary, gave a classification (up to equivariant homeomorphism) of all compact Lie group actions - except the circle - on 3-manifolds. Orlik and Raymond completed the picture in 1968 by classifying all circle actions on closed 3-manifolds. More recently, attention has shifted to actions of homeomorphism groups. Despite the deep differences from Lie groups, Militon (2017) classified all actions of Homeo$_0(S^1)$ on the annulus and torus. Chen and Mann (2019) extended this, classifying Homeo$_0(S^1)$-actions on all compact surfaces and developing general tools to study such actions. Using these tools, this dissertation proves a Homeo$_0(M)$ tube theorem and a weak orbit type stratification for actions of manifold homeomorphism groups, offering structural results in analogy with classical Lie group actions.