Fixed-point sets of semifree group actions on infinite-dimensional compact convex sets and their manifolds
Abstract
Let A be the fixed point set of a semifree action of a compact (metric) group G on a Keller space X (an infinite-dimensional compact convex subset of a separable Fr\' echet space). All Keller spaces are homeomorphic and all nontrivial compact Lie groups act on all X's semifreely with unique fixed points (see Section 3). We prove (1) if A is a single point, then it is the fixed point set of a semifree action of G with uniformly small orbits, (2) if A has Property Z (e.g., is of infinite codimension) in X, then the action of G may be replaced by one with fixed point set A and uniformly small orbits, and (3) if A is an \ANR, then G acts on X semifreely with fixed point set homeomorphic to any compact (metric) space that is Shape equivalent to A. Corollaries are (a) every Cell-like set (i.e., continuum of trivial Shape) embedded in a Keller space X as a Z-set is the fixed point set of semifree actions on X with uniformly small orbits of all nontrivial compact Lie groups (Corollary \ref{CE}) and (b) (a partial converse to a Theorem of P.\ A.\ Smith) if a compact (metric) space Y has the Shape of a compact \ANR, then Y embeds in X as the fixed point set of a homeomorphism of X of period n if it is acyclic in \v Cech homology with ℤn coefficients. Finally, we (4) apply recent results of Cappell, Weinberger, and Yan \cite{cwy} to give for a compact manifold M modeled on a Keller space homological and algebraic K-theoretic conditions equivalent to the existence of semifree actions of finite groups G on M with prescribed \ANR\ fixed point sets and (5) generalize (3) to prove that if A is an \ANR\ that is the fixed point set of a semifree action of G on M, then every compact (metric) space that is Shape equivalent to A is homeomorphic to the fixed point set of a semifree action of G on M.
Related
- Classification of analytic SOo(p,q)-actions on closed (p+q−1)-dimensional manifolds II : p ≥ 3, q=1,2
- Boundary Dynamics, Cubical Actions, and Infinite Girth
- Finite group actions on nonpositively curved 3-manifolds
- Group actions on von Neumann algebras with compact open subgroups
- Self-intersecting sections of polar actions on simply connected manifolds
BibTeX
@misc{mill2026fixedpoint,
title = {{Fixed-point sets of semifree group actions on infinite-dimensional compact convex sets and their manifolds}},
author = {Jan van Mill and James E. West},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09561},
eprint = {2610.09561},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09561},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fixed-point sets of semifree group actions on infinite-dimensional compact convex sets and their manifolds
AU - Jan van Mill
AU - James E. West
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09561
UR - https://arxiv.org/abs/2610.09561
AB - Let A be the fixed point set of a semifree action of a compact (metric) group G on a Keller space X (an infinite-dimensional compact convex subset of a separable Fr\' echet space). All Keller spaces are homeomorphic and all nontrivial compact Lie groups act on all X's semifreely with unique fixed points (see Section 3). We prove (1) if A is a single point, then it is the fixed point set of a semifree action of G with uniformly small orbits, (2) if A has Property Z (e.g., is of infinite codimension) in X, then the action of G may be replaced by one with fixed point set A and uniformly small orbits, and (3) if A is an \ANR, then G acts on X semifreely with fixed point set homeomorphic to any compact (metric) space that is Shape equivalent to A. Corollaries are (a) every Cell-like set (i.e., continuum of trivial Shape) embedded in a Keller space X as a Z-set is the fixed point set of semifree actions on X with uniformly small orbits of all nontrivial compact Lie groups (Corollary \ref{CE}) and (b) (a partial converse to a Theorem of P.\ A.\ Smith) if a compact (metric) space Y has the Shape of a compact \ANR, then Y embeds in X as the fixed point set of a homeomorphism of X of period n if it is acyclic in \v Cech homology with ℤ_n coefficients. Finally, we (4) apply recent results of Cappell, Weinberger, and Yan \cite{cwy} to give for a compact manifold M modeled on a Keller space homological and algebraic K-theoretic conditions equivalent to the existence of semifree actions of finite groups G on M with prescribed \ANR\ fixed point sets and (5) generalize (3) to prove that if A is an \ANR\ that is the fixed point set of a semifree action of G on M, then every compact (metric) space that is Shape equivalent to A is homeomorphic to the fixed point set of a semifree action of G on M.
ER -