Realization problems for diffeomorphism
                    groups

Mann, Kathryn; Tshishiku, Bena

doi:10.1090/pspum/102/11

Cited by 15 publications

(18 citation statements)

References 44 publications

(59 reference statements)

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“…All manifolds, for the remainder of the paper, will be assumed smooth and orientable. We assume basic familiarity with classifying spaces for topological groups, the reader may refer to [MT18] for a very brief introduction in the context of related section problems for diffeomorphism groups, or [Mor01] for more detailed background.…”

Section: Obstruction Classes and A Proof When M Is Irreduciblementioning

confidence: 99%

Dynamical and cohomological obstructions to extending group actions

Mann

Nariman

2020

Math. Ann.

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We study cohomological obstructions to extending group actions on the boundary ∂M of a 3-manifold to a C 0 -action on M when ∂M is diffeomorphic to a torus or a sphere. In particular, we show that for a 3-manifold M with torus boundary which is not diffeomorphic to a solid torus, the torus action on the boundary does not extend to a C 0 -action on M . Moreover, we use techniques from 3-manifold topology, homotopy theory, and low-dimensional dynamics to find group actions on a torus and a sphere that are not nullbordant.

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Section: Obstruction Classes and A Proof When M Is Irreduciblementioning

confidence: 99%

Dynamical and cohomological obstructions to extending group actions

Mann

Nariman

2020

Math. Ann.

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“…This problem presents two kinds of difficulties: the lack of understanding of finite index subgroups of Mod(S g ) and the lack of understanding of relations in Homeo + (S g ). To illustrate the latter, we state the following problem [15]. Problem 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…This problem presents two kinds of difficulties: the lack of understanding of finite index subgroups of

Mod (S_{g})

and the lack of understanding of relations in

{prefixHomeo}_{+} false(S_{g} false)

. To illustrate the latter, we state the following problem [15]. Problem Give an example of a finitely generated, torsion‐free group

normalΓ

, and a surface S, such that

normalΓ

is not isomorphic to a subgroup of

{prefixHomeo}_{+} false(S false)

.…”

Section: Introductionmentioning

confidence: 99%

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Non‐realizability of the Torelli group as area‐preserving homeomorphisms

Chen

Marković

2020

J. London Math. Soc.

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Nielsen realization problem for the mapping class group Mod(Sg) asks whether the natural projection pg : Homeo+(Sg) → Mod(Sg) has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion-free subgroups of Mod(Sg). The main result of this paper is that the Torelli group has no realization inside the area-preserving homeomorphisms.

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“…The Nielsen realization problem for P B n is widely open since the two methods in [12] and [1] fail to work and have no hope to repair. The following question is asked by [8,Question 3.12] and [12,Remark 1.4]. PROBLEM 1.1.…”

Section: Introductionmentioning

confidence: 99%

Non-realizability of the pure braid group as area-preserving homeomorphisms

Chen

2020

Ergod. Th. Dynam. Sys.

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Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$ . All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$ , the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

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Realization problems for diffeomorphism groups

Cited by 15 publications

References 44 publications

Dynamical and cohomological obstructions to extending group actions

Dynamical and cohomological obstructions to extending group actions

Non‐realizability of the Torelli group as area‐preserving homeomorphisms

Non-realizability of the pure braid group as area-preserving homeomorphisms

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