3-manifold groups

Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry

doi:10.48550/arxiv.1205.0202

Cited by 26 publications

(52 citation statements)

References 389 publications

(494 reference statements)

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“…Substituting all the SL(2, C) integrations in (33) with SU (2) integrations as in (35), substituting formulae ( 34) and (36) in the result and performing all of the integrations over u and v using equation (40), we find:…”

Section: Calculation Of Transition Amplitudes and Probabilitiesmentioning

confidence: 99%

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Including topology change in Loop Quantum Gravity with topspin network formalism with application to homogeneous and isotropic cosmology

Villani

2021

Preprint

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We apply topspin network formalism to Loop Quantum Gravity in order to include in the theory the possibility of changes in the topology of spacetime. We apply this formalism to three toy models: with the first, we find that the topology can actually change due to the action of the Hamiltonian constraint and with the second we find that the final state might be a superposition of states with different topologies. In the third and last application, we consider an homogeneous and isotropic Universe, calculating the difference equation that describes the evolution of the system and which are the final topological states after the action of the Hamiltonian constraint. For this last case, we also calculate the transition amplitudes and probabilities from the initial to the final states.

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Section: Calculation Of Transition Amplitudes and Probabilitiesmentioning

confidence: 99%

“…We follow [23] and use Fox free calculus [24][25][26][27][28][29][30][31]. With this method, one can calculate the fundamental group of the manifold which is a topological invariant and can be used to classify 3-manifolds [32,33]. If it changes during the evolution of the system it means that the topology of the manifold has changed.…”

Section: Introductionmentioning

confidence: 99%

Including topology change in Loop Quantum Gravity with topspin network formalism with application to homogeneous and isotropic cosmology

Villani

2021

Preprint

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“…I expect the result to be generalizable to the non-orientable case by the standard trick of working in the orientable double cover, but I haven't checked all details. By prime decomposition theorem (see [25] for a brief review of the subject and [26] which is more up to date), orientable compact 3d manifolds can be decomposed as a connected sum of prime manifolds…”

Section: Some Facts About 3d Topologymentioning

confidence: 99%

“…N is locally maximized on γ. 16 To derive (26) note that a tangent η µ to a slice V u is Lie-transported along the foliation:…”

Section: Iv) Mcf Avoids Crushing Singularities (Theorem 4)mentioning

confidence: 99%

Topology of Cosmological Black Holes

Mirbabayi

2018

Preprint

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Motivated by the question of how generic inflation is, I study the time-evolution of topological surfaces in an inhomogeneous cosmology with positive cosmological constant Λ. If matter fields satisfy the Weak Energy Condition, non-spherical incompressible surfaces of least area are shown to expand at least exponentially, with rate d log A min /dλ ≥ 8πG N Λ, under the mean curvature flow parametrized by λ. With reasonable assumptions about the nature of singularities this restricts the topology of black holes: (a) no trapped surface or apparent horizon can be a non-spherical, incompressible surface, and (b) the interior of black holes cannot contain any such surface.

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“…In the hyperbolic case this theorem is due to Wise [Wi12], in the graph manifold case it is due to Liu [Liu11] and independently to Przytycki-Wise [PW11] and in the remaining ('mixed') case it is due to Przytycki-Wise [PW12]. We refer to the survey paper [AFW12] for background and more precise references. 4.2.…”

Section: Proof Of the 'Only If' Direction Of Theorem 11mentioning

confidence: 99%