Journal of Topology - current issue

Criteria for virtual fibering

Agol, I. — 2008-04-17

We prove that an irreducible 3-manifold with fundamental group that satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber dodecahedral space and the Bianchi groups. Moreover, we show that a taut-sutured compression body has a finite-sheeted cover with a depth one taut-oriented foliation.

Cobordism of knots on surfaces

Turaev, V. — 2008-04-17

We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.

The integral Novikov conjectures for linear groups containing torsion elements

Ji, L. — 2008-04-17

In this paper, we show that for any global field k, the generalized integral Novikov conjecture in both K- and L-theories holds for every finitely generated subgroup of GL(n, k). This implies that the conjecture holds for every finitely generated subgroup of , where is the algebraic closure of . We also show that for every linear algebraic group defined over k, every S-arithmetic subgroup satisfies this generalized integral Novikov conjecture. We note that the integral Novikov conjecture implies the stable Borel conjecture, in particular, the stable Borel conjecture holds for all the above torsion-free groups. Most of these subgroups are not discrete subgroups of Lie groups with finitely many connected components, and some of them are not finitely generated. When the field k is a function field such as , and the k-rank of is positive, many of these S-arithmetic subgroups such as do not admit cofinite universal spaces for proper actions.

Operations on the A-theoretic nil-terms

Grunewald, J., Klein, J. R., Macko, T. — 2008-04-17

For a space X, we define Frobenius and Verschiebung operations on the nil-terms in the algebraic K-theory of spaces, in three different ways. Two applications are included. Firstly, we show that the homotopy groups of are either trivial or not finitely generated as abelian groups. Secondly, the Verschiebung operation defines a -module structure on the homotopy groups of , with the multiplicative monoid.

We also give a calculation of the homotopy type of the nil-terms after p-completion for an odd prime p and their homotopy groups as -modules up to dimension 4p – 7. We obtain non-trivial groups only in dimension 2p – 2, where it is finitely generated as a -module, and in dimension 2p – 1, where it is not finitely generated as a -module.

Dimension of asymptotic cones of Lie groups

de Cornulier, Y. — 2008-04-17

We compute the covering dimension of the asymptotic cone of a connected Lie group. For simply connected solvable Lie groups, this is the codimension of the exponential radical.

As an application of the proof, we give a characterization of connected Lie groups that quasi-isometrically embed into a nonpositively curved metric space.

Projection genericity of space curves

Wall, C. T. C. — 2008-04-17

If is a smooth space curve, we consider the family of projections of from a variable point not on to a fixed plane. For a residual set of curves , this family versally unfolds those singularities that occur in it.

To obtain a family of curves which is open as well as dense in the space of smooth maps, we must compactify the parameter space, so we study curves in real projective space, and include projections from points of the curve itself. If is smoothly embedded, the projection C_P of from is a well-defined smooth curve, and for generic the family C_P has generic singularities.

However, when the point of projection moves off , the projection varies discontinuously. We define a family of plane curves, parametrised by the blow-up X of P³ along , such that for a point in the exceptional locus lying over , we have the union of the projection C_P of from P and a straight line L through the image of the tangent at P. A key result asserts that this is a flat family. We give an explicit list of restrictions on the family C_P L (the key condition is that the total contact order of C_P with L never exceeds 2), and show that these hold for a dense open set of curves , and that if they do hold, there is a neighbourhood U of , such that the family of projections from points of is generic.

Combining this list of conditions with those obtained previously gives a natural definition of a dense set of space curves , for which the complete family of projections has generic singularities, and we show that this set is also open.

The homotopy invariance of the string topology loop product and string bracket

Cohen, R. L., Klein, J. R., Sullivan, D. — 2008-04-17

Let Mⁿ be a closed, oriented, n-manifold, and LM its free loop space. In [Chas and Sullivan, ‘String topology’, Ann. of Math., to appear] a commutative algebra structure in homology, H_*(LM), and a Lie algebra structure in equivariant homology , were defined. In this paper, we prove that these structures are homotopy invariants in the following sense. Let f:M₁ -> M₂ be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equivalence, Lf:LM₁ -> LM₂ induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds true for any generalized homology theory h_* that supports an orientation of the M_i.

Constructing infinitely many smooth structures on small 4-manifolds

Akhmedov, A., Baykur, R. I., Park, B. D. — 2008-04-17

The purpose of this article is two-fold. First we outline a general construction scheme for producing simply connected minimal symplectic -manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic -manifolds homeomorphic but not diffeomorphic to for k = 1, ..., 4, or to for l = 1, ..., 6. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on, and .

Mirror symmetry formulae for the elliptic genus of complete intersections

Gorbounov, V., Ochanine, S. — 2008-04-17

In this paper, we calculate the elliptic genus of certain complete intersections in products of projective spaces. We show that it is equal to the elliptic genus of the Landau–Ginzburg models that are, according to Hori and Vafa, mirror partners of these complete intersections. This provides additional evidence of the validity of their construction.

On the topological classification of certain singular hypersurfaces in 4-dimensional projective space

Su, Y. — 2008-04-17

In this paper, the classification of hypersurfaces in with an isolated singularity are studied. If the singularity is of type A_k, under certain restrictions of the degree of the hypersurfaces, a classification up to homeomorphism, which is a diffeomorphism on the nonsingular part, is obtained. Examples of cubic hypersurfaces with an A₅-singularity are constructed.

The first homology of the group of equivariant diffeomorphisms and its applications

Abe, K., Fukui, K. — 2008-04-17

Let V be a representation space of a finite group G. We determine the group structure of the first homology of the equivariant diffeomorphism group of V. Then we can apply it to the calculation of the first homology of the corresponding automorphism groups of smooth orbifolds, compact Hausdorff foliations, codimension one or two compact foliations and the locally free S¹-actions on 3-manifolds.

Rational blowdowns and smoothings of surface singularities

Stipsicz, A. I., Szabo, Z., Wahl, J. — 2008-04-17

In this paper, we give a necessary combinatorial condition for a negative-definite plumbing tree to be suitable for rational blowdown, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are also presented; these include singularities with resolution graph having valency 4 nodes.

Chern numbers and diffeomorphism types of projective varieties

Kotschick, D. — 2008-04-17

In 1954, Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. We give a complete answer to this question in small dimensions, and also prove partial results without restrictions on the dimension.