Cusps of hyperbolic 4‐manifolds and rational homology spheres

Ferrari, Leonardo; Kolpakov, Alexander; Slavich, Leone

doi:10.1112/plms.12421

Cited by 7 publications

(10 citation statements)

References 19 publications

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“…If we use the more general notion of colouring of Remark 4, nontoric cusps may also appear (see, for instance, [15]).…”

Section: Remarkmentioning

confidence: 99%

“…Example 11. If we consider P = P 8 with its 15-colouring, there are 2 15 vertices in C, and 16 edges connecting v to v + e j for every v and every j.…”

Section: Diagonal Mapsmentioning

confidence: 99%

“…Remark 19. By analogy with the 24-cell, it would be tempting to guess that the 2 15 states in the orbit are all isomorphic, and maybe that the ascending and descending links are homotopic to two copies of S 3 linked in S 7 . This is, however, impossible by a Euler characteristic argument, due to the fact that the 24-cell has χ = 1, while χ(P 8 ) = 17/2 is much bigger.…”

Section: A 1-legal Orbit For Pmentioning

confidence: 99%

“…This produces a hyperbolic 8-manifold M 8 , tessellated into 2 15 = 32,768 copies of P 8 . We have χ(M 8 ) = 2 15 • 17/2 = 27,8528.…”

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confidence: 99%

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Hyperbolic Manifolds That Fibre Algebraically Up to Dimension 8

Italiano

Martelli²,

Migliorini³

2022

J. Inst. Math. Jussieu

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We construct some cusped finite-volume hyperbolic n-manifolds $M^n$ that fibre algebraically in all the dimensions $5\leq n \leq 8$ . That is, there is a surjective homomorphism $\pi _1(M^n) \to {\mathbb {Z}}$ with finitely generated kernel. The kernel is also finitely presented in the dimensions $n=7, 8$ , and this leads to the first examples of hyperbolic n-manifolds $\widetilde M^n$ whose fundamental group is finitely presented but not of finite type. These n-manifolds $\widetilde M^n$ have infinitely many cusps of maximal rank and, hence, infinite Betti number $b_{n-1}$ . They cover the finite-volume manifold $M^n$ . We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes $P^5, \ldots , P^8$ , and then applying some arguments of Jankiewicz, Norin and Wise [18] and Bestvina and Brady [7]. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to $P^n$ , and the algebra of integral octonions for the crucial dimensions $n=7,8$ .

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“…If we use the more general notion of colouring of Remark 4, nontoric cusps may also appear (see, for instance, [15]).…”

Section: Remarkmentioning

confidence: 99%

“…Example 11. If we consider P = P 8 with its 15-colouring, there are 2 15 vertices in C, and 16 edges connecting v to v + e j for every v and every j.…”

Section: Diagonal Mapsmentioning

confidence: 99%

Section: A 1-legal Orbit For Pmentioning

confidence: 99%

“…This produces a hyperbolic 8-manifold M 8 , tessellated into 2 15 = 32,768 copies of P 8 . We have χ(M 8 ) = 2 15 • 17/2 = 27,8528.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Hyperbolic Manifolds That Fibre Algebraically Up to Dimension 8

Italiano

Martelli²,

Migliorini³

2022

J. Inst. Math. Jussieu

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“…In this section we briefly describe a well-known generalisation of the notion of colouring (see [FKS21] for a complete discussion). Let W be a compact n-manifold with corners.…”

Section: Generalised Colouringsmentioning

confidence: 99%

Dodecahedral L-spaces and hyperbolic 4-manifolds

Battista¹,

Ferrari²,

Santoro³

2022

Preprint

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We prove that exactly 6 out of the 29 rational homology 3-spheres tessellated by four or less right-angled hyperbolic dodecahedra are L-spaces. The algorithm used is based on the L-space census provided by Dunfield in [Dun20], and relies on a result by Rasmussen-Rasmussen [RR17]. We use the existence of these manifolds together with a result of Martelli [Mar16a] to construct explicit examples of hyperbolic 4-manifolds containing separating L-spaces, and therefore having vanishing Seiberg-Witten invariants. This answers a question asked by Agol and Lin in [AL20].

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Infinitesimal rigidity for cubulated manifolds

Battista

2023

Geom Dedicata

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We prove the infinitesimal rigidity of some geometrically infinite hyperbolic 4- and 5-manifolds. These examples arise as infinite cyclic coverings of finite-volume hyperbolic manifolds obtained by colouring right-angled polytopes, already described in the papers (Battista in Trans Am Math Soc 375(04):2597–2625, 2022; Italiano et al. in Invent Math, 2022. https://doi.org/10.1007/s00222-022-01141-w). The 5-dimensional example is diffeomorphic to $$N\times {{\mathbb {R}}}$$ N × R for some aspherical 4-manifold N which does not admit any hyperbolic structure. To this purpose, we develop a general strategy to study the infinitesimal rigidity of cyclic coverings of manifolds obtained by colouring right-angled polytopes.

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Cusps of hyperbolic 4‐manifolds and rational homology spheres

Cited by 7 publications

References 19 publications

Hyperbolic Manifolds That Fibre Algebraically Up to Dimension 8

Hyperbolic Manifolds That Fibre Algebraically Up to Dimension 8

Dodecahedral L-spaces and hyperbolic 4-manifolds

Infinitesimal rigidity for cubulated manifolds

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