Journal of Topology - recent issues

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.

A new twist on Lorenz links

Birman, J., Kofman, I. — Fri, 10 Jul 2009 06:41:52 PDT

Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T-links are a natural generalization given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T-links, so Lorenz links and T-links coincide. Using this correspondence, we identify over half of the simplest hyperbolic knots as Lorenz knots. We show that both hyperbolic volume and the Mahler measure of Jones polynomials are bounded for infinite collections of hyperbolic Lorenz links. The correspondence provides unexpected symmetries for both Lorenz links and T-links, and establishes many new results for T-links, including new braid index formulas.

Simply connected asymmetric manifolds

Kreck, M. — Fri, 10 Jul 2009 06:41:52 PDT

In this paper, we give the first examples of smooth simply connected asymmetric closed manifolds. V. Puppe has shown that there are examples of simply connected 6-manifolds on which there is no orientation-preserving nontrivial group action of a finite group. We show that all of them are actually asymmetric manifolds. The main tool in the proof is a congruence of a twisted -genus for certain Spin-manifolds admitting an orientation-reversing involution. This is the first restriction of this type coming from an action of a discrete group.

Brown representability follows from Rosicky's theorem

Neeman, A. — Fri, 10 Jul 2009 06:41:52 PDT

We prove that the dual of a well-generated triangulated category satisfies Brown representability, as long as there is a combinatorial model. This settles the major open problem in [12]. We also prove that Brown representability holds for non-dualized well-generated categories, but that only amounts to the fourth known proof of the fact. The proof depends crucially on a new result of J. Rosicky, Theory and Applications of Categories 14 (2005) no. 19, 451–479.

On the K-theory of truncated polynomial algebras over the integers

Angeltveit, V., Gerhardt, T., Hesselholt, L. — Fri, 10 Jul 2009 06:41:52 PDT

We show that is finite of order (mi)!(i!)^m–2 and that is free abelian of rank m – 1. This is accomplished by showing that the equivariant homotopy groups of the topological Hochschild -spectrum are free abelian for q even, and finite for q odd, and by determining their ranks and orders, respectively.

On {beta}-elements in the Adams-Novikov spectral sequence

Nakai, H., Ravenel, D. C. — Fri, 10 Jul 2009 06:41:52 PDT

In this paper we detect invariants in the comodule consisting of β-elements over the Hopf algebroid (A(m+1), G(m+1)) defined in [4], and we show that some related Ext groups vanish below a certain dimension. The result obtained here will be extensively used in future work with Ravenel to extend the range of our knowledge for _*(T(m)) obtained in [4].

The refined transfer, bundle structures, and algebraic K-theory

Klein, J. R., Williams, B. — Fri, 10 Jul 2009 06:41:52 PDT

We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibres admits a reduction to a fibre bundle with compact topological manifold fibres. The criteria lead to an unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen's functor A(X). We also give concrete examples of fibrations having a reduction to a fibre bundle with compact topological manifold fibres but which fail to admit a compact fibre smoothing. The examples are detected by algebraic K-theory invariants. We consider a refinement of the Becker–Gottlieb transfer. We show that a version of the axioms described by Becker and Schultz uniquely determines the refined transfer for the class of fibrations, admitting a reduction to a fibre bundle with compact topological manifold fibres. In the Appendix, we sketch a theory of characteristic classes for fibrations. The classes are primary obstructions to finding a compact fibre smoothing.

The norm residue isomorphism theorem

Weibel, C. — Fri, 10 Jul 2009 06:41:52 PDT

We provide a patch to complete the proof of the Voevodsky–Rost Theorem, that the norm residue map is an isomorphism. (This settles the motivic Bloch–Kato conjecture.)

Bers slices are Zariski dense

Dumas, D., Kent, R. P. — Fri, 10 Jul 2009 06:41:52 PDT

We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.

Floer homology and singular knots

Ozsvath, P., Stipsicz, A., Szabo, Z. — Fri, 10 Jul 2009 06:41:52 PDT

In this paper we define and investigate variants of the link Floer homology introduced by the first and third authors. More precisely, we define Floer homology theories for oriented, singular knots in S³ and show that one of these theories can be calculated combinatorially for planar singular knots.

Smoothing nodal Calabi-Yau n-folds

Rollenske, S., Thomas, R. — Fri, 10 Jul 2009 06:41:52 PDT

Let X be an n-dimensional Calabi–Yau with ordinary double points, where n is odd. Friedman showed that for n = 3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also true, due to work of Friedman, Kawamata, and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n > 3 the result is non-linear; the `Yukawa product' on the middle-dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi–Yau hypersurfaces of .

Symplectic embeddings of 4-dimensional ellipsoids

McDuff, D. — Thu, 26 Mar 2009 11:14:28 PDT

We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into CP². For example, the problem of embedding the ellipsoid E(1, k) into a ball B is equivalent to that of embedding k disjoint equal balls into CP², and so can be solved by the work of Gromov, McDuff–Polterovich, and Biran. (Here k is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid E(1, k) for k = 1, 4 and all k >= 9, thus answering a question raised by Hofer.

Ribbon R-trees and holomorphic dynamics on the unit disk

McMullen, C. T. — Thu, 26 Mar 2009 11:14:28 PDT

Let C denote the unit disk, viewed as a model for the hyperbolic plane. Under rescaling, takes on the appearance of a tree, with an additional ribbon structure coming from the cyclic ordering of its ends.

In this paper, we show that branched coverings of ribbon trees naturally compactify the space of proper holomorphic maps f : (, 0) -> (, 0), and use the structure of these ribbon trees to describe the limiting moduli of f.

The almost alternating diagrams of the trivial knot

Tsukamoto, T. — Thu, 26 Mar 2009 11:14:28 PDT

Bankwitz characterized the alternating diagrams of the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize the almost alternating diagrams of the trivial knot. As a corollary, we determine the unknotting number one alternating knots with the property that the unknotting operation can be done on its alternating diagram.

Handle moves in contact surgery diagrams

Ding, F., Geiges, H. — Thu, 26 Mar 2009 11:14:28 PDT

We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the classification of loose Legendrian knots. Along the way, we prove a one-to-one correspondence (up to Legendrian isotopy) between long Legendrian knots in 3-space and their completion in the 3-sphere.

Equivariant representable K-theory

Emerson, H., Meyer, R. — Thu, 26 Mar 2009 11:14:28 PDT

We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups and compute these via a classifying space and as K-theory groups of suitable -C^*-algebras. We also relate equivariant vector bundles to these -C^*-algebras and provide sufficient conditions for equivariant vector bundles to generate representable K-theory. We mostly work in the generality of locally compact groupoids with Haar systems.

Minimal triangulations for an infinite family of lens spaces

Jaco, W., Rubinstein, H., Tillmann, S. — Thu, 26 Mar 2009 11:14:28 PDT

The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein, and unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its minimal layered triangulation. This paper proves that for each n >= 2, the minimal layered triangulation of the lens space L(2n, 1) is its unique minimal triangulation. More generally, the minimal triangulations (and hence the complexity) are determined for an infinite family of lens spaces containing the lens space of the form L(2n, 1).

Minimum volume hyperbolic 3-manifolds

Milley, P. — Thu, 26 Mar 2009 11:14:28 PDT

We enumerate the small-volume manifolds that can be obtained by Dehn filling on Mom-2 and Mom-3 manifolds as defined by Gabai, Meyerhoff, and the author. In so doing we complete the proof that the Weeks manifold is the compact hyperbolic 3-manifold of minimum volume, as well as enumerating the ten smallest one-cusped hyperbolic 3-manifolds.

Degree theorems and Lipschitz simplicial volume for nonpositively curved manifolds of finite volume

Loh, C., Sauer, R. — Thu, 26 Mar 2009 11:14:28 PDT

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively curved manifolds or locally symmetric spaces of noncompact type. In contrast, we provide vanishing results for the ordinary simplicial volume; for instance, we show that the ordinary simplicial volume of noncompact locally symmetric spaces with finite volume of Q-rank at least 3 is zero.