{
  "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 $\\mathbb{Z}_{n}$ 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 $M$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09561",
  "author_total": 2,
  "authors": [
    {
      "name": "Jan van Mill"
    },
    {
      "name": "James E. West"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7f5ay894j1q4f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09561",
  "fulltext_url": null,
  "id": "7f5ay894j1q4f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09561",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7220989424",
  "pdf_url": "https://arxiv.org/pdf/2610.09561",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7f5ay894j1q4f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09561",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09561"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Fixed-point sets of semifree group actions on infinite-dimensional compact convex sets and their manifolds",
  "title_text": "Fixed-point sets of semifree group actions on infinite-dimensional compact convex sets and their manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/7f5ay894j1q4f",
  "venue": "arXiv",
  "versions": []
}