On 4-Dimensional 2-Handlebodies and 3-Manifolds

Bobtcheva, Ivelina; Piergallini, Riccardo

doi:10.1142/s0218216512501106

Cited by 13 publications

(66 citation statements)

References 49 publications

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“…In fact, it is satisfied by any branched covering representation of such a 3‐manifold derived from an integral surgery description of it by the procedure given in Montesinos (cf. also Edmonds ) or by the more effective one provided in Bobtcheva and Piergallini (see Section 3). Theorem Every compact connected oriented PL 4‐manifold

M

with

n

boundary components can be represented by a simple branched covering

p : M \to S^{4} - Int false(B_{1}^{4} \cup \dots \cup B_{n}^{4} false)

satisfying property

a

b

as in Theorem , with the PL 4‐balls

B_{i}^{4}

disjointly and canonically embedded in

S^{4}

and

B_{p}

a bounded surface properly immersed or embedded in

S^{4} - Int false(B_{1}^{4} \cup \dots \cup B_{n}^{4} false)

.…”

Section: Definitions and Results In The Pl Categorymentioning

confidence: 99%

“…Our starting point is the branched covering representation of compact connected oriented 4‐dimensional 2‐handlebodies up to 2‐deformations that is provided in Bobtcheva and Piergallini . As usual, here and in the following, we call a 2‐handlebody any handlebody whose handles all have index at most 2, and a 2‐deformation any sequence of handle operations (isotopy, sliding and addition/deletion of canceling handles) not involving any handle of index

>

2.…”

Section: Proofsmentioning

confidence: 99%

“…Below we briefly recall the procedure described in [, Section 3] (see also [, Sections 6.1 and 3.4]) for deriving from any Kirby diagram

K

of a connected oriented 4‐dimensional 2‐handlebody

H

a labeled ribbon surface

S_{K} \subset B^{4}

representing a simple 3‐fold covering

p : H \to B^{4}

branched over

S_{K}

.…”

Section: Proofsmentioning

confidence: 99%

“…The construction above depends on various choices, the significant ones being in steps 1, 5, 6 and 7. However, the labeled ribbon surfaces obtained from different choices become equivalent up to labeled isotopy of ribbon surfaces in

B^{4}

(called 1‐isotopy in Bobtcheva and Piergallini ) and the covering moves

R_{1}

and

R_{2}

depicted in Figure , after adding to them a separate trivial disk with label

(3 4)

. We recall that the addition of such disk represents the stabilization of the branched covering

p

with an extra trivial fourth sheet to give a simple 4‐fold branched covering

\tilde{p} : H ≅ H #_{\partial} B^{4} \to B^{4}

.…”

Section: Proofsmentioning

confidence: 99%

“…The aim of the present article is to answer these questions. This can be done in light of the results obtained by Bobtcheva and the first author in (see also ), about covering moves relating different branched coverings of

B^{4}

having PL homeomorphic covering spaces.…”

Section: Introductionmentioning

confidence: 99%

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On branched covering representation of 4‐manifolds

Piergallini

Zuddas

2018

J. London Math. Soc.

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Assuming M to be a connected oriented PL 4-manifold, our main results are the following:(1) if M is compact with (possibly empty) boundary, there exists a simple branched coveringIn both cases, the degree d(p) and the branching set Bp of p can be assumed to satisfy one of these conditions: (1) d(p) = 4 and Bp is a properly self-transversally immersed locally flat PL surface; (2) d(p) = 5 and Bp is a properly embedded locally flat PL surface. In the compact (respectively, open) case, by relaxing the assumption on the degree we can have B 4 (respectively, R 4 ) as the base of the covering.A crucial technical tool used in all the proofs is a quite delicate cobordism lemma for coverings of S 3 , which also allows us to obtain a relative version of the branched covering representation of bounded 4-manifolds, where the restriction to the boundary is a given branched covering.We also define the notion of branched covering between topological manifolds, which extends the usual one in the PL category. In this setting, as an interesting consequence of the above results, we prove that any closed oriented topological 4-manifold is a 4-fold branched covering of S 4 . According to almost smoothability of 4-manifolds, this branched covering could be wild at a single point.

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M

with

n

boundary components can be represented by a simple branched covering

p : M \to S^{4} - Int false(B_{1}^{4} \cup \dots \cup B_{n}^{4} false)

satisfying property

a

b

as in Theorem , with the PL 4‐balls

B_{i}^{4}

disjointly and canonically embedded in

S^{4}

and

B_{p}

a bounded surface properly immersed or embedded in

S^{4} - Int false(B_{1}^{4} \cup \dots \cup B_{n}^{4} false)

.…”

Section: Definitions and Results In The Pl Categorymentioning

confidence: 99%

>

2.…”

Section: Proofsmentioning

confidence: 99%

“…Below we briefly recall the procedure described in [, Section 3] (see also [, Sections 6.1 and 3.4]) for deriving from any Kirby diagram

K

of a connected oriented 4‐dimensional 2‐handlebody

H

a labeled ribbon surface

S_{K} \subset B^{4}

representing a simple 3‐fold covering

p : H \to B^{4}

branched over

S_{K}

.…”

Section: Proofsmentioning

confidence: 99%

B^{4}

(called 1‐isotopy in Bobtcheva and Piergallini ) and the covering moves

R_{1}

and

R_{2}

depicted in Figure , after adding to them a separate trivial disk with label

(3 4)

. We recall that the addition of such disk represents the stabilization of the branched covering

p

with an extra trivial fourth sheet to give a simple 4‐fold branched covering

\tilde{p} : H ≅ H #_{\partial} B^{4} \to B^{4}

.…”

Section: Proofsmentioning

confidence: 99%

B^{4}

having PL homeomorphic covering spaces.…”

Section: Introductionmentioning

confidence: 99%

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On branched covering representation of 4‐manifolds

Piergallini

Zuddas

2018

J. London Math. Soc.

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Spin foams and noncommutative geometry

Denicola

Marcolli

Al-Yasry

2010

Class. Quantum Grav.

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We extend the formalism of embedded spin networks and spin foams to include topological data that encode the underlying three-manifold or four-manifold as a branched cover. These data are expressed as monodromies, in a way similar to the encoding of the gravitational field via holonomies. We then describe convolution algebras of spin networks and spin foams, based on the different ways in which the same topology can be realized as a branched covering via covering moves, and on possible composition operations on spin foams. We illustrate the case of the groupoid algebra of the equivalence relation determined by covering moves and a 2-semigroupoid algebra arising from a 2-category of spin foams with composition operations corresponding to a fibered product of the branched coverings and the gluing of cobordisms. The spin foam amplitudes then give rise to dynamical flows on these algebras, and the existence of low temperature equilibrium states of Gibbs form is related to questions on the existence of topological invariants of embedded graphs and embedded two-complexes with given properties. We end by sketching a possible approach to combining the spin network and spin foam formalism with matter within the framework of spectral triples in noncommutative geometry. Contents

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Topspin networks in loop quantum gravity

Duston¹

2012

Class. Quantum Grav.

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We discuss the extension of loop quantum gravity to topspin networks, a proposal which allows topological information to be encoded in spin networks. We will show that this requires minimal changes to the phase space, C*-algebra and Hilbert space of cylindrical functions. We will also discuss the area and Hamiltonian operators, and show how they depend on the topology. This extends the idea of "background independence" in loop quantum gravity to include topology as well as geometry. It is hoped this work will confirm the usefulness of the topspin network formalism and open up several new avenues for research into quantum gravity. PACS numbers: 04.60.-m,04.60.Pp I σ I J σ −1 J = 1, (1.4)for incoming arcs I and outgoing arcs J. These relations encode how the sheets of the covering are stitched together so that π 1 (S 3 \ Γ) is appropriately represented. The idea of a topspin network is to identify the branched covers, decorated with permutation labels σ I under the Wirtinger relations, with the usual spin networks of LQG, ensuring that the spin and topological labels are compatible. In this way the spin

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On 4-Dimensional 2-Handlebodies and 3-Manifolds

Cited by 13 publications

References 49 publications

On branched covering representation of 4‐manifolds

On branched covering representation of 4‐manifolds

Spin foams and noncommutative geometry

Topspin networks in loop quantum gravity

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