Journal of Topology - current issue

Alexander-equivalent Zariski pairs of irreducible sextics

Eyral, C., Oka, M. — Mon, 05 Oct 2009 22:18:04 PDT

The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 was proved by Degtyarev. However, no explicit example of such a pair is available (only the existence is known) in the literature. In this paper, we construct the first concrete example.

The conjugacy problem in subgroups of right-angled Artin groups

Crisp, J., Godelle, E., Wiest, B. — Mon, 05 Oct 2009 22:18:04 PDT

We prove that the conjugacy problem in a large and natural class of subgroups of right-angled Artin groups can be solved in linear time. This class of subgroups has been previously studied by Crisp and Wiest, and independently by Haglund and Wise, as fundamental groups of compact special cube complexes.

Derivatives of embedding functors I: the stable case

Arone, G. — Mon, 05 Oct 2009 22:18:04 PDT

For smooth manifolds M and N, let be the homotopy fibre of the inclusion map . Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula . In this paper, we describe the derivatives of this functor, in the sense of M. Weiss's orthogonal calculus, in the case when N is a stably parallelizable manifold (we believe that the parallelizability assumption is not essential). Our construction involves a certain space of partitions of M (or, equivalently, a space of rooted forests with leaves marked by points in M), and a certain `homotopy bundle of spectra' over this space of trees. The nth derivative is then described as the `spectrum of restricted sections' of this bundle.

On 'maximal' poles of zeta functions, roots of b-functions, and monodromy Jordan blocks

Melle-Hernandez, A., Torrelli, T., Veys, W. — Mon, 05 Oct 2009 22:18:04 PDT

The main objects of this study are the poles of several local zeta functions: the Igusa, topological, and motivic zeta function associated to a polynomial or (germ of) holomorphic function in n variables. We are interested in poles of maximal possible order n. In all known cases (curves, non-degenerate polynomials) there is at most one pole of maximal order n, which is then given by the log canonical threshold of the function at the corresponding singular point. For an isolated singular point we prove that if the log canonical threshold yields a pole of order n of the corresponding (local) zeta function, then it induces a root of the Bernstein–Sato polynomial of the given function of multiplicity n (proving one of the cases of the strongest form of a conjecture of Igusa–Denef–Loeser). For an arbitrary singular point, we show under the same assumption that the monodromy eigenvalue induced by the pole has ‘a Jordan block of size n on the (perverse) complex of nearby cycles’.

Symplectic Jacobi diagrams and the Lie algebra of homology cylinders

Habiro, K., Massuyeau, G. — Mon, 05 Oct 2009 22:18:04 PDT

Let S be a compact connected oriented surface whose boundary is connected or empty. A homology cylinder over the surface S is a cobordism between S and itself, homologically equivalent to the cylinder over S. The Y-filtration on the monoid of homology cylinders over S is defined by clasper surgery. Using a functorial extension of the Le–Murakami–Ohtsuki invariant, we show that the graded Lie algebra associated to the Y-filtration is isomorphic to the Lie algebra of ‘symplectic Jacobi diagrams’. This Lie algebra consists of the primitive elements of a certain Hopf algebra whose multiplication is a diagrammatic analogue of the Moyal–Weyl product. The mapping cylinder construction embeds the Torelli group into the monoid of homology cylinders, sending the lower central series to the Y-filtration. We give a combinatorial description of the graded Lie algebra map induced by this embedding, by connecting Hain’s infinitesimal presentation of the Torelli group to the Lie algebra of symplectic Jacobi diagrams. This Lie algebra map is shown to be injective in degree 2, and the question of the injectivity in higher degrees is discussed.

The space of closed subgroups of is stratified and simply connected

Kloeckner, B. — Mon, 05 Oct 2009 22:18:04 PDT

The Chabauty space of a topological group is the set of its closed subgroups, endowed with a natural topology. As soon as n > 2, the Chabauty space of has a rather intricate topology and is not a manifold. By an investigation of its local structure, we fit it into a wider, but not too wild, class of topological spaces (namely, Goresky–MacPherson stratified spaces). Thanks to a localization theorem, this local study also leads to the main result of this article: the Chabauty space of is simply connected for all n.

Some 6-dimensional Hamiltonian S1-manifolds

McDuff, D. — Mon, 05 Oct 2009 22:18:04 PDT

In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into by using a new way to desingularize orbifold blow-ups Z of the weighted projective space . We now use a related method to construct symplectomorphisms of these spaces Z. This allows us to construct some well-known Fano 3-folds (including the Mukai–Umemura 3-fold) in purely symplectic terms using a classification by Tolman of a particular class of Hamiltonian S¹-manifolds. We also show that (modulo scaling) these manifolds are uniquely determined by their fixed-point data up to equivariant symplectomorphism. As part of this argument, we show that the symplectomorphism group of a certain weighted blow-up of a weighted projective plane is connected.

On Kontsevich's characteristic classes for higher-dimensional sphere bundles II: Higher classes

Watanabe, T. — Mon, 05 Oct 2009 22:18:04 PDT

This paper studies Kontsevich’s characteristic classes of smooth bundles with fibre in a ‘singularly framed’ odd-dimensional homology sphere, which are defined through his graph complex and configuration space integral. We will give a systematic construction of smooth bundles parameterized by trivalent graphs and will show that our smooth bundles are non-trivially detected by Kontsevich’s characteristic classes. It turns out that there are surprisingly many non-trivial elements of the rational homotopy groups of the diffeomorphism groups of spheres that are in some ‘non-stable’ range. In particular, the homotopy groups of the diffeomorphism groups in some ‘non-stable’ range are not finite.