Ekaterina Pervova

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.

COMPLEXITY OF LINKS IN 3-MANIFOLDS

We introduce a complexity c(X) ∈ ℕ for pairs X = (M,L), where M is a closed orientable 3-manifold and L ⊂ M is a link. The definition employs simple spines, but for well-behaved Xs, we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveevs recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0, 2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0, 2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.

Diffeological Clifford algebras and pseudo-bundles of Clifford modules

Abstract We consider the diffeological version of the Clifford algebra of a diffeological finite dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and considering which of its standard properties re-appear in the diffeological context (most of them), we turn to our main interest, which is constructing the pseudo-bundles of Clifford algebras associated to a given (finite dimensional) diffeological vector pseudo-bundle, and those of the usual Clifford modules (the exterior algebras). The substantial difference that emerges with respect to the standard context, and paves the way to various questions that do not have standard analogues, stems from the fact that the notion of a diffeological pseudo-bundle is very different from the usual bundle, and this under two main respects: it may have fibres of different dimensions, and even if it does not, its total and base spaces frequently are not smooth, or even topological, manifolds.

Pseudo-bundles of Exterior Algebras as Diffeological Clifford Modules

We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring that the dual of the result of gluing together two pseudo-bundles can equivalently be obtained by gluing together their duals, which is not automatic in the diffeological context. We show that, assuming that the dual of the gluing map, which itself does not have to be a diffeomorphism, on the total space is one, the commutativity condition is satisfied, via a natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved.

Generalized Mom-structures and ideal triangulations of 3–manifolds with nonspherical boundary

The so-called Mom-structures on hyperbolic cusped 3-manifolds without boundary were introduced by Gabai, Meyerhoff, and Milley, and used by them to identify the smallest closed hyperbolic manifold. In this work we extend the notion of a Mom-structure to include the case of 3-manifolds with non-empty boundary that does not have spherical components. We then describe a certain relation between such generalized Mom-structures, called protoMom-structures, internal on a fixed 3-manifold N, and ideal triangulations of N; in addition, in the case of non-closed hyperbolic manifolds without annular cusps, we describe how an internal geometric protoMom-structure can be constructed starting from Epstein-Penner or Kojima decomposition. Finally, we exhibit a set of combinatorial moves that relate any two internal protoMom-structures on a fixed N to each other.

NOTES ON THE COMPLEXITY OF 3-VALENT GRAPHS IN 3-MANIFOLDS

A theory of complexity for pairs (M, G) with M an arbitrary closed 3-manifold and G ⊂ M a 3-valent graph was introduced by the first two named authors, extending the original notion due to Matveev. The complexity c is known to be always additive under connected sum away from the graphs, but not always under connected sum along (unknotted) arcs of the graphs. In this article, we prove the slightly surprising fact that if in M there is a sphere intersecting G transversely at one point, and this point belongs to an edge e of G, then e can be canceled from G without affecting the complexity. Using this fact we completely characterize the circumstances under which complexity is additive under connected sum along graphs. For the set of pairs (M, K) with K ⊂ M a knot, we also prove that any function that is fully additive under connected sum along knots is actually a function of the ambient manifold only.