We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of B 4 whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition.

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We show that for any n ≥ 4 there exists an equivalence functor [Formula: see text] from the category [Formula: see text] of n-fold connected simple coverings of B3 × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category [Formula: see text] of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S3 branched over links, which provides a complete solution to the long-standing Fox–Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S3 branched over embedded graphs. Then, we factor the functor above as [Formula: see text], where [Formula: see text] is an equivalence functor to a universal braided category [Formula: see text] freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category [Formula: see text]. From this we derive an analogous description of the category [Formula: see text] of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

We prove the long-standing Montesinos conjecture that any closed oriented PL 4-manifold M is a simple covering of S 4 branched over a locally flat surface (cf [12]). In fact, we show how to eliminate all the node singularities of the branching set of any simple 4-fold branched covering M → S 4 arising from the representation theorem given in [13]. Namely, we construct a suitable cobordism between the 5-fold stabilization of such a covering (obtained by adding a fifth trivial sheet) and a new 5-fold covering M → S 4 whose branching set is locally flat. It is still an open question whether the fifth sheet is really needed or not.

Analogue gravitational systems are becoming an increasing popular way of studying the behaviour of quantum systems in curved spacetime. Setups based on ultracold quantum gases in particular, have been recently harnessed to explore the thermal nature of Hawking's and Unruh's radiation that was theoretically predicted almost 50 years ago. For solid state implementations, a promising system is graphene, in which a link between the Dirac-like low-energy electronic excitations and relativistic quantum field theories has been unveiled soon after its discovery. This link could be extended to the case of curved quantum field theory when the graphene sheet is shaped in a surface of constant negative curvature, known as Beltrami's pseudosphere.Here we provide numerical evidence that energetically stable negative curvature graphene surfaces can be realized. Owing to large-scale simulations, our geometrical realizations are characterised by a ratio between the carbon-carbon bond length and the pseudosphere radius small enough to allow the formation of an analog of a black hole event horizon. Additionally, from the energy dependence of the spatially resolved density of states, we infer some thermal properties of the corresponding gravitational system, which could be investigated using low temperature scanning tunnelling microscopy or optical near field spectroscopy. These findings pave the way to the realization of a solid-state system in which the curved spacetime dynamics of quantum many body systems can be investigated.Quantum mechanics and general relativity are the most successful theories of