Journal of Topology - Advance Access

The simplicial volume of closed manifolds covered by

Bucher-Karlsson, M. — 2008-05-15

We compute the value of the simplicial volume for closed, oriented Riemannian manifolds covered by explicitly, and thus in particular for products of closed hyperbolic surfaces. This gives the first exact value of a nonvanishing simplicial volume for a manifold not admitting a hyperbolic structure.

The structure and singularities of quotient arc complexes

Penner, R. C. — 2008-05-12

A well-known combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is an analogous complex QA(F) consisting of equivalence classes of arcs in F connecting a given finite set of points in its boundary modulo diffeomorphisms of F pointwise fixing the boundary and any punctures. The main result of this paper is the determination of those complexes QA(F) that are also spheres. This classification has consequences for Riemann's moduli space of curves via its known identification with a related quotient arc complex in the punctured case with no boundary. Namely, the essential singularities of the natural cellular compactification of Riemann's moduli space can be described.

On cyclic branched coverings of prime knots

Boileau, M., Paoluzzi, L. — 2008-05-09

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non-equivalent to K. To prove the main theorem, a result concerning symmetries of knots is also obtained. This latter result can be interpreted as a characterisation of the trivial knot.

Leray numbers of projections and a topological Helly-type theorem

Kalai, G., Meshulam, R. — 2008-04-28

Let X be a simplicial complex on the vertex set V. The rational Leray number of X is the minimal d, such that for all induced subcomplexes Y X and i >= d. Suppose that is a partition of V such that the induced subcomplexes X[V_i] are all 0-dimensional. Let denote the projection of X into the (m – 1)-simplex on the vertex set {1, ..., m} given by (v) = i if v V_i. Let r = max{|^–1((x))|:x |X|}. It is shown that One consequence is a topological extension of a Helly-type result of Amenta. Let be a family of compact sets in such that for any , the intersection is either empty or contractible. It is shown that if is a family of sets such that for any finite , the intersection is a union of at most r disjoint sets in , then the Helly number of is at most r(d + 1).