Finite rigid sets in sphere complexes

A subcomplex X≤C of a simplicial complex is strongly rigid if every locally injective, simplicial map X→C is the restriction of a unique automorphism of C. Aramayona and the second author proved that the curve complex of an orientable surface can be exhausted by finite strongly rigid sets. The Hatch...

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Bibliographic Details
Published inTopology and its applications Vol. 347; p. 108862
Main Authors Bering, Edgar A., Leininger, Christopher J.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 15.04.2024
Subjects
Outer automorphism group
Rigidity
Sphere complex
20F65
57M50
Outer automorphism group
Rigidity
05C25
20E36
Sphere complex
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Summary:A subcomplex X≤C of a simplicial complex is strongly rigid if every locally injective, simplicial map X→C is the restriction of a unique automorphism of C. Aramayona and the second author proved that the curve complex of an orientable surface can be exhausted by finite strongly rigid sets. The Hatcher sphere complex is an analog of the curve complex for isotopy classes of essential spheres in a connect sum of n copies of S1×S2. We show that there is an exhaustion of the sphere complex by finite strongly rigid sets for all n≥3 and that when n=2 the sphere complex does not have finite rigid sets.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2024.108862