Riccardo Benedetti

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.

Cosmological time in (2+1)-gravity

Abstract We consider maximal globally hyperbolic flat (2+1)-spacetimes with compact space S of genus g>1. For any spacetime M of this type, the length of time that the events have been in existence is M defines a global time, called the cosmological time CT of M, which reveals deep intrinsic properties of spacetime. In particular, the past/future asymptotic states of the cosmological time recover and decouple the linear and the translational parts of the ISO(2,1)-valued holonomy of the flat spacetime. The initial singularity can be interpreted as an isometric action of the fundamental group of S on a suitable real tree. The initial singularity faithfully manifests itself as a lack of smoothness of the embedding of the CT level surfaces into the spacetime M. The cosmological time determines a real analytic curve in the Teichmuller space of Riemann surfaces of genus g, which connects an interior point (associated to the linear part of the holonomy) with a point on Thurstons natural boundary (associated to the initial singularity).

Classical and quantum dilogarithmic invariants of flat PSL(2,C) -bundles over 3-manifolds

We introduce a family of matrix dilogarithms, which are automorphisms of C N ⊗ C N , N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 2 → 3 move on 3–dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N = 1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3–manifolds W endowed with a flat principal P SL(2, C)–bundle ρ, and a fixed non empty link L if N > 1, and for (possibly “marked”) cusped hyperbolic 3–manifolds M. When N = 1 the state sums recover known simplicial formulas for the volume and the Chern–Simons invariant. When N > 1, the invariants for M are new; those for triples (W, L, ρ) coincide with the quantum hyperbolic invariants defined in [3], though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N = 1 and N > 1, and we formulate “Volume Conjectures”, having geometric motivations, about the asymptotic behaviour of the invariants when N → ∞. AMS Classification numbers Primary: 57M27, 57Q15 Secondary: 57R20, 20G42

Bounding the number of connected components of a real algebraic set

For every polynomial mapf=(f1,…,fk): ℝn→ℝk, we consider the number of connected components of its zero set,B(Zf) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree offi), and thek-tuple (Δ1,...,Δ4), Δk being the Newton polyhedron offi respectively. Our aim is to boundB(Zf) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μd(n)=d(2d−1)n−1. Considered as a polynomial ind, μd(n) has leading coefficient equal to 2n−1. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μd(n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)nk−1dn, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thoms argument, Smiths theory, and information about the sum of Betti numbers of complex complete intersections.

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.

The topology of two-dimensional real algebraic varieties

SuntoÈ noto che ogni spazio analitico reale è localmente omeomorfo al cono su un poliedro con caratteristica di Eulero-Poincaré pari. Si dimostra che questa condizione è anche sufficiente affinchè un poliedro (compatto) di dimensione due P sia omeomorfo ad una varietà algebrica reale affine P. Segue inoltre dalla costruzione che la P ottenuta ha, in un certo senso, un insieme di singolarità algebriche minimale, compatibilmente con la topologia di P.

Spin and pin− structures, immersed and embedded surfaces and a result of Segre on real cubic surfaces

THE initial motivation of this paper comes from a result of Segre [ 121 about the real lines on a real cubic surface. As it is well known a smooth complex cubic surface has exactly 27 (complex) lines. In the real case this is not always true anymore. A smooth real cubic surface can have 27,15,7 or 3 real lines. This has been well known since the 19th century. The result of Segre we are alluding to is far less known and introduces a more subtle difference between the real and complex cases. Segre distinguishes two types of real straight lines (see below Section 6 for precise definitions) and shows that on a real cubic surface with 27 real lines 15 are of one type and 12 of the other (in fact the result is more complete and gives the classification in all cases-see Theorem 6.2 below). Segre proved this result by studying the degeneration of non-singular cubic surfaces to the union of 3 planes and a special “graphical” way of representing the occurring situations. Noting that the basic difference between the two types of lines is that their respective tubular neighbourhoods in the surface differ by a full twist in P3, our initial aim was to give a new interpretation and a new proof of this result in terms of the Pinstructure induced by the embedding of the surface in P3 (p3 taken with a fixed Spin structure). More precisely, we will show that the two type of lines distinguished by Segre are also differentiated at the homology level by the mod4 quadratic form canonically associated with the above Pinstructure. A further point of interest is that, assuming that the complexification X(C) c p’(C) of the surface X is also non-singular and that the surface is an M-surface, there is another Pinstructure, induced by the embedding of X(R) in X(@) (see [6]). This second form differs from the first by a “privileged” class in H’ (X, Z/2), a class which seems to deserve further investigations. We will explicitly compute this class for quadric and cubic surfaces. The work done for surfaces in P3 led us to study more generally immersions of surfaces in arbitrary orientable 3-manifolds. Using the theory of Spin and Pinstructures (see, for example, [lo] and the book [6]), we consider the problem of associating, as above, quadratic forms with immersions of surfaces in 3-manifolds. We have done this by reformulating results of Pinkall [ 111, where only the case of Iw3 is considered and results of Hass and Hughes [S] where the immersions of surfaces into arbitrary 3-manifolds is studied, but not in terms of quadratic forms. Following Pinkall we will also introduce the notion of immersed surfaces (an equivalence class of immersions-see Sections 4 and 10) and study the relationships between different equivalence relations on immersed surfaces (regular homotoppy, cobordism, equivalence of the Pinstructures) extending the results Pinkall

On simultaneous diagonalization of one Hermitian and one symmetric form

Abstract It is remarked that if A , B ϵ M n ( C ), A = A t , and B = B t , B positive definite, there exists a nonsingular matrix U such that (1) Ū t BU = I and (2) U t AU is a diagonal matrix with nonnegative entries. Some related actions of the real orthogonal group and equations involving the unitary group are studied.

Geometric Cone Surfaces and (2+1) - Gravity coupled to Particles

We introduce the ( 2+1 )-spacetimes with compact space of genus g≥0 and r gravitating particles which arise by three kinds of construction called: (a) the Minkowskian suspension of flat or hyperbolic cone surfaces; (b) the distinguished deformation of hyperbolic suspensions; (c) the patchworking of suspensions. Similarly to the matter-free case, these spacetimes have nice properties with respect to the canonical Cosmological Time Function. When the values of the masses are sufficiently large and the cone points are suitably spaced, the distinguished deformations of hyperbolic suspensions determine a relevant open subset of the full parameter space; this open subset is homeomorphic to U×R6g−6+2r , where U is a non empty open set of the Teichmuller space Trg . By patchworking of suspensions one can produce examples of spacetimes which are not distinguished deformations of any hyperbolic suspensions, although they have the same topology and same masses; in fact, we will guess that they belong to different connected components of the parameter space.