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Automatic continuity for homeomorphism groups and applications

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Kathryn Mann

Geometric Topology · Submitted 12 Jan 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous.

how it is stated · asserted 1.00 read the record →

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arXiv ID1501.02688 (version 2)
Pith ID6AWP6IRG, Pith's permanent record of this paper and its review
CitationKathryn Mann. "Automatic continuity for homeomorphism groups and applications." arXiv:1501.02688 (2015). https://pith.science/paper/6AWP6IRG
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,831 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.12 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.21 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, the group of homeomorphisms of M that preserve N also has this property. Various applications of automatic continuity are discussed, including applications to the topology and structure of groups of germs of homeomorphisms. In an appendix with Frederic Le Roux we also show, using related techniques, that the group of germs at a point of homeomorphisms of Rn is strongly uniformly simple.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let M be a compact manifold, possibly with boundary. background 0.98 0.08 0.01 asserted 0.68
s1 We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. result 0.93 0.97 0.94 asserted 1.00
s2 This answers a question of C. Rosendal. interpretation 0.37 0.39 0.06 asserted 0.97
s3 If N is a submanifold of M, the group of homeomorphisms of M that preserve N also has this property. result 0.99 0.92 0.05 asserted 1.00
s4 Various applications of automatic continuity are discussed, including applications to the topology and structure of groups of germs of homeomorphisms. contribution 0.75 0.79 0.04 asserted 0.96
s5 In an appendix with Frederic Le Roux we also show, using related techniques, that the group of germs at a point of homeomorphisms of Rn is strongly uniformly simple. result 0.83 0.95 0.03 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • references and citing sentences
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  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, the group of homeomorphisms of M that preserve N also has this property. Various applications of automatic continuity are discussed, including applications to the topology and structure of groups of germs of homeomorphisms. In an appendix with Frederic Le Roux we also show, using related techniques, that the group of germs at a point of homeomorphisms of Rn is strongly uniformly simple.

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How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper
  • Other instances of this algebraic–topological relationship can be seen in the main results of [ref] , [ref] , [ref] , and [ref] .

    PL(M) admits no Polish group topology · Introduction · 0.93

What it cites · 2 sentences without a confident label

Cites as background 1 paper
  • In [ref] , Hurtado defines a notion of weak continuity for homomorphisms – a homomorphism $ϕ$ from $_0(M)$ to another group is weakly continuous if, for every compact set $K ⊂M$, the restriction of $ϕ$ to the subgroup of homeomorphisms with support contained in $K$ is continuous.

    Continuity of discrete homomorphisms of diffeomorphism groups · Noncompact manifolds · 0.82

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