Quantitative boundedness in homeomorphism groups

We obtain sharp boundedness exponents from finite-correction obstructions and continuously varying geometric normal forms. With the compact-open topology, the homeomorphism group of the two-sphere and the full and orientation-preserving homeomorphism groups of planar telescoping surfaces have least open-exhaustion exponent five and least finite-neighbourhood exponent three. The same pair holds for continuous-function groups from nonempty nondiscrete locally compact second-countable Hausdorff spaces into the boundary-fixing homeomorphism group of a ball of any positive dimension. Five-factor bounds for further geometric targets refine known qualitative boundedness arguments; the full-sphere bound and the pointwise boundary-fixing ball word already occur in published proofs. A controlled Hilbert-cube argument supplies a quantitative bound and repairs the earlier qualitative proof. We also extend ultralocal boundedness from closed compact manifolds to compact manifolds with boundary.

This paper develops material from the author's monograph Property (OB) and the Coarse Geometry of Polish Groups. It is one of eight separate papers intended to replace the monograph.

GitHub repository: https://github.com/eulogica/property-ob-independent-papers