Carlo Petronio

Feynman diagrams of generalized matrix models and the associated manifolds in dimension four

The problem of constructing a quantum theory of gravity has been tackled with very different strategies, most of which rely on the interplay between ideas from physics and from advanced mathematics. On the mathematical side, a central role is played by combinatorial topology, often used to recover the space–time manifold from the other structures involved. An extremely attractive possibility is that of encoding all possible space–times as specific Feynman diagrams of a suitable field theory. In this work we analyze how exactly one can associate combinatorial four-manifolds with the Feynman diagrams of certain tensor theories.

Three-manifolds having complexity at most 9

We describe an algorithm which has enabled us to give a complete list, without repetitions, of all closed oriented irreducible three-manifolds of complexity up to 9. More interestingly, we have actually been able to give a name to each such manifold, by recognizing its canonical decomposition into Seifert fibered spaces and hyperbolic manifolds. The algorithm relies on the extension of Matveevs theory of complexity to the caseof manifolds bounded by suitably marked tori, and on the notion of assembling of two such manifolds. We show that every manifold is an assembling of manifolds which cannot be further disassembled, and we prove that there are surprisingly few such manifolds up to complexity 9. Our most interesting experimental discovery is that there are 4 closed hyperbolic manifolds having complexity 9, and they are the 4 closed hyperbolic manifolds of least known volume. All other manifolds having complexity at most 9 are graph manifolds.

Branched Standard Spines of 3-Manifolds

Motivations, plan and statements.- A review on standard spines and o-graphs.- Branched standard spines.- Manifolds with boundary.- Combed closed manifolds.- More on combings, and the closed calculus.- Framed and spin manifolds.- Branched spines and quantum invariants.- Problems and perspectives.- Homology and cohomology computations.

TWO-SIDED ASYMPTOTIC BOUNDS FOR THE COMPLEXITY OF SOME CLOSED HYPERBOLIC THREE-MANIFOLDS

We establish two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Lobell manifolds and the Fibonacci manifolds.

Small hyperbolic 3-manifolds with geodesic boundary

We classify the orientable finite-volume hyperbolic 3-manifolds having nonempty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, describe their canonical Kojima decomposition, and discuss manifolds having cusps. The eight manifolds built from one or two tetrahedra were previously known. There are 151 different manifolds built from three tetrahedra, realizing 18 different volumes. Their Kojima decomposition always consists of tetrahedra (but occasionally requires four of them). There is a single cusped manifold, which we can show to be a knot complement in a genus-2 handlebody. Concerning manifolds built from four tetrahedra, we show that there are 5,033 different ones, with 262 different volumes. The Kojima decomposition consists either of tetrahedra (as many as eight of them in some cases), of two pyramids, or of a single octahedron. There are 30 manifolds having a single cusp and one having two cusps. Our results were obtained with the aid of a computer. The complete list of manifolds (in SnapPea format) and full details on their invariants are available on the world wide web.

COMBED 3-MANIFOLDS WITH CONCAVE BOUNDARY, FRAMED LINKS, AND PSEUDO-LEGENDRIAN LINKS

We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M, v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly @M 6 ∅, and v may be tangent to @M, but only in a concave fashion, and homotopy should preserve tangency type; (2) framed links L in M, up to framed isotopy; (3) triples (M, v, L), with (M, v) as above and L transversal to v, up to pseudo-Legendrian isotopy (transversality- preserving simultaneous homotopy of v and isotopy of L). All realizations are based on the notion of branched standard spine, and build on results previously obtained. Links are encoded by means of diagrams on branched spines, where the diagram is C 1 with respect to the branching. Several motivations for being interested in combinatorial realizations of the topological categories considered in this paper are given in the introduction. The encoding of links is suitable for the comparison of the framed and the pseudo-Legendrian categories, and some applications are given in connection with contact structures, torsion and finite-order invariants. An estension of Traces notion of winding number of a knot diagram is introduced and discussed.

ON ROBERTS’ PROOF OF THE TURAEV-WALKER THEOREM

In this paper we discuss the beautiful idea of Justin Roberts [7] (see also [8]) to re-obtain the Turaev-Viro invariants [11] via skein theory, and re-prove elementarily the Turaev-Walker theorem [9], [10], [13]. We do this by exploiting the presentation of 3-manifolds introduced in [1], [4]. Our presentation supports in a very natural way a formal implementation of Roberts’ idea. More specifically, what we show is how to explicitly extract from an o-graph (the object by which we represent a manifold, see below), one of the framed links in S3 which Roberts uses in the construction of his invariant, and a planar diagrammatic representation of such a link. This implies that the proofs of invariance and equality with the Turaev-Viro invariant can be carried out in a completely “algebraic” way, in terms of a planar diagrammatic calculus which does not require any interpretation of 3-dimensional figures. In particular, when proving the “term-by-term” equality of the expansion of the Roberts invariant with the state sum which gives the Turaev-Viro invariant, we simultaneously apply several times the “fusion rule” (which is formally defined, strictly speaking, only in diagrammatic terms), showing that the “braiding and twisting” which a priori may exist on tetrahedra is globally dispensable. In our point of view the success of this formal “algebraic” approach witnesses a certain efficiency of our presentation of 3-manifolds via o-graphs. In this work we will widely use recoupling theory which was very clearly exposed in [2], and therefore we will avoid recalling notations. Actually, for the purpose of stating and proving our results we will need to slightly extend the class of trivalent ribbon diagrams on which the bracket can be computed. We also address the reader to the references quoted in [2], in particular for the fundamental contributions of Lickorish to this area. In our approach it is more natural to consider invariants of compact 3-manifolds with non-empty boundary. The case of closed 3-manifolds is included by introducing a correction factor corresponding to boundary spheres, as explained in §2. Our main result is actually an extension to manifolds with boundary of the Turaev-Walker theorem: we show that the Turaev-Viro invariant of such a manifold coincides (up to a factor which depends on the Euler characteristic) with the Reshetikhin-Turaev-Witten invariant of the manifold mirrored in its boundary.

ON THE EXISTENCE OF BRANCHED COVERINGS BETWEEN SURFACES WITH PRESCRIBED BRANCH DATA, II

If is a branched covering between closed surfaces, there are several easy relations one can establish between the Euler characteristics and χ(Σ), orientability of Σ and , the total degree, and the local degrees at the branching points, including the classical Riemann–Hurwitz formula. These necessary relations have been shown to be also sufficient for the existence of the covering except when Σ is the sphere 𝕊 (and when Σ is the projective plane, but this case reduces to the case Σ = 𝕊). For Σ = 𝕊 many exceptions are known to occur and the problem is widely open. Generalizing methods of Baranski, we prove in this paper that the necessary relations are actually sufficient in a specific but rather interesting situation. Namely under the assumption that Σ = 𝕊, that there are three branching points, that one of these branching points has only two pre-images with one being a double point, and either that and that the degree is odd, or that has genus at least one, with a single specific exception. For the case of we also show that for each even degree there are precisely two exceptions.

Spherical splitting of 3-orbifolds

The famous Haken–Kneser–Milnor theorem states that every 3-manifold can be expressed in a unique way as a connected sum of prime 3-manifolds. The analogous statement for 3-orbifolds has been part of the folklore for several years, and it was commonly believed that slight variations on the argument used for manifolds would be sufficient to establish it. We demonstrate in this paper that this is not the case, proving that the apparently natural notion of “essential” system of spherical 2-orbifolds is not adequate in this context. We also show that the statement itself of the theorem must be given in a substantially different way. We then prove the theorem in full detail, using a certain notion of “efficient splitting system.”

Surface branched covers and geometric 2-orbifolds

Let Σ and Σ be closed, connected, and orientable surfaces, and let f: Σ → Σ be a branched cover. For each branching point x ∈ Σ the set of local degrees of f at f ―1 (x) is a partition of the total degree d. The total length of the various partitions is determined by χ(Σ), χ(Σ), d and the number of branching points via the Riemann-Hurwitz formula. A very old problem asks whether a collection of partitions of d having the appropriate total length (that we call a candidate cover) always comes from some branched cover. The answer is known to be in the affirmative whenever Σ is not the 2-sphere S, while for Σ = S exceptions do occur. A long-standing conjecture however asserts that when the degree d is a prime number a candidate cover is always realizable. In this paper we analyze the question from the point of view of the geometry of 2-orbifolds, and we provide strong supporting evidence for the conjecture. In particular, we exhibit three different sequences of candidate covers, indexed by their degree, such that for each sequence: • The degrees giving realizable covers have asymptotically zero density in the naturals. • Each prime degree gives a realizable cover.