Finiteness of mapping degree sets for 3-manifolds

Derbez, Pierre; Sun, Hongbin; Wang, Shi Cheng

doi:10.1007/s10114-011-0416-x

Cited by 12 publications

(8 citation statements)

References 13 publications

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“…We will now focus on the 3-dimensional case. The classification of weakly flexible 3-manifolds by Derbez, Sun, and Wang [3] translates into the following result: Corollary 3.9. Let M be an oriented closed connected 3-manifold.…”

Section: Secondary Simplicial Volumementioning

confidence: 96%

Exotic Finite Functorial Semi-Norms on Singular Homology

Fauser¹,

Löh²

2018

Glasgow Math. J.

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Section: Secondary Simplicial Volumementioning

confidence: 96%

Exotic Finite Functorial Semi-Norms on Singular Homology

Fauser¹,

Löh²

2018

Glasgow Math. J.

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“…This invariant has been introduced and studied by R. Brooks and W. Goldman [BG1,BG2,Go] as a geometrical analogue of the celebrated simplicial volume of orientable closed manifolds due to M. Gromov [Gr,Th1]. During the past few years, much has been known about the ( SL 2 (R) × Z R)representation volume (the Seifert volume) and the PSL(2, C)-representation volume (the hyperbolic volume) for 3-manifolds and their finite covers [DW1,DW2,DLW,DSW,DLSW]. Those invariants have demonstrated to be useful in studying nonzero degree maps between 3-manifolds, especially when the simplicial volume vanishes.…”

Section: Introductionmentioning

confidence: 99%

Volume of representations and mapping degree

Derbez

Liu

Sun

et al. 2019

Advances in Mathematics

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Given a connected real Lie group and a contractible homogeneous proper Gspace X furnished with a G-invariant volume form, a real valued volume can be assigned to any representation ρ : π 1 (M ) → G for any oriented closed smooth manifold M of the same dimension as X. Suppose that G contains a closed and cocompact semisimple subgroup, it is shown in this paper that the set of volumes is finite for any given M . From a perspective of model geometries, examples are investigated and applications with mapping degrees are discussed.

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“…Let D be the set of all closed orientable 3-manifolds N with D(M, N ) finite for any fixed M . By Theorem 1.1 (4), SV(N ) = HV(N ) = 0 if N / ∈ D. It is known that (see [7], for example) N ∈ D if and only if N contains a prime factor Q with non-trivial geometric decomposition, or supporting an SL 2 (R) or a hyperbolic geometry. This fact combined with Theorem 1.1(2), (3), (4) implies that if vol(N, Iso e SL 2 (R)) = {0}, then necessarily a prime factor of N has a non-trivial geometric decomposition, or supports an SL 2 (R) or a hyperbolic geometry and if vol(N, PSL(2; C)) = {0}, then necessarily a prime factor of N contains some hyperbolic JSJ pieces.…”

Section: Introductionmentioning

confidence: 99%

Chern–Simons theory, surface separability, and volumes of 3-manifolds

2015

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We study the set vol(M, G) of volumes of all representations ρ : π1M → G, where M is a closed oriented 3-manifold and G is either Iso+H 3 or Isoe SL2(R).By various methods, including relations between the volume of representations and the Chern-Simons invariants of flat connections, and recent results of surfaces in 3-manifolds, we prove that any 3-manifold M with positive Gromov simplicial volume has a finite cover M with vol( M, Iso+H 3 ) = {0}, and that any non-geometric 3-manifold M containing at least one Seifert piece has a finite cover M with vol( M, Isoe SL2(R)) = {0}.We also find 3-manifolds M with positive simplicial volume but vol(M, Iso+H 3 ) = {0}, and non-trivial graph manifolds M with vol(M, Isoe SL2(R)) = {0}, proving that it is in general necessary to pass to some finite covering to guarantee that vol(M, G) = {0}.Besides we determine vol(M, G) when M supports the Seifert geometry. ContentsDefinition 4.9. Let M be an orientable closed irreducible mixed 3-manifold containing no essential Klein bottles. Let J 0 be a JSJ piece, T 0 be a JSJ torus adjacent to J 0 and ζ 0 be a slope on T 0 . A partial PW subsurface j : R M is said to be virtually bounded by ζ 0 outside J 0 if the boundary ∂R of R is non-empty, covering ζ 0 under j, and if the interiorR of R misses J 0 under j. In this case, the carrier chunk X(R) ⊂ M of R is the unique minimal chunk that contains R, and the carrier boundary of X(R) is the component T 0 ⊂ ∂X(R).Definition 4.10. We say that a partial PW subsurface j : R M is parallel cutting if, for every JSJ torus T ⊂ M , all components of j −1 (T ) in R cover the same slope of T . Virtual existence of partial PW subsurfacesTheorem 4.11. Let M be an orientable closed irreducible mixed 3-manifold containing no essential Klein bottles. Let ζ 0 be a slope on a JSJ torus T 0 adjacent to a JSJ piece J 0 . Then, for some finite coverM of M together with an elevation (J 0 ,T 0 ,ζ 0 ) of the triple (J 0 , T 0 , ζ 0 ), Pierre Derbez I2M UMR 7373

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Finiteness of mapping degree sets for 3-manifolds

Abstract: By constructing certain maps, this note completes the answer of the Question: For which closed orientable 3-manifold N , the set of mapping degrees D(M, N ) is finite for any closed orientable 3-manifold M ? Date: June 9, 2018. 1991 Mathematics Subject Classification. 57M99, 55M25.

Cited by 12 publications

References 13 publications

Exotic Finite Functorial Semi-Norms on Singular Homology

Exotic Finite Functorial Semi-Norms on Singular Homology

Volume of representations and mapping degree

Chern–Simons theory, surface separability, and volumes of 3-manifolds

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