Journal of Topology - recent issues

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.

Surgery on nullhomologous tori and simply connected 4-manifolds with b+ = 1

Fintushel, R., Stern, R. J. — 2007-12-17

For 5 ≤ k ≤ 8, we show that an infinite family of exotic smooth structures on CP²#kCP² can be obtained by 1/n-surgeries on a single embedded nullhomologous torus in a manifold R_k which is homeomorphic to CP²#kCP².

Twisted equivariant K-theory with complex coefficients

Freed, D. S., Hopkins, M. J., Teleman, C. — 2007-12-17

Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact group acting on itself by conjugation and relate the result to the Verlinde algebra and to the Kac numerator at q=1. Verlinde's formula is also discussed in this context.

Axiomatic characterization of ordinary differential cohomology

Simons, J., Sullivan, D. — 2007-12-17

The Cheeger–Simons differential characters, the Deligne cohomology in the smooth category, the Hopkins–Singer construction of ordinary differential cohomology, and the recent Harvey–Lawson constructions are each in two distinct ways abelian group extensions of known functors. In one description, these objects are extensions of integral cohomology by the quotient space of all differential forms by the subspace of closed forms with integral periods. In the other, they are extensions of closed differential forms with integral periods by the cohomology with coefficients in the circle. These two series of short-exact sequences mesh with two interlocking long-exact sequences (the Bockstein sequence and the de Rham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Our first theorem shows that on the category of smooth manifolds and smooth maps, any package consisting of a functor into graded abelian groups together with four natural transformations that fit together so as to form a Character Diagram as mentioned earlier is unique up to a unique natural equivalence. Our second theorem shows that natural product structure on differential characters is uniquely characterized by its compatibility with the product structures on the known functors in the Character Diagram. The proof of our first theorem couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.

On the Farrell Jones Conjecture and its applications

Bartels, A., Luck, W., Reich, H. — 2007-12-17

We present the status of the Farrell–Jones Conjecture for algebraic K-theory for a group G and arbitrary coefficient rings R. We add new groups for which the conjecture is known to be true, and we study inheritance properties. We discuss new applications, focussing on the Bass Conjecture, the Kaplansky Conjecture, and conjectures generalizing Moody's Induction Theorem. Thus, we considerably extend the class of groups for which these conjectures are known.

The tower of K-theory of truncated polynomial algebras

Hesselholt, L. — 2007-12-17

Let A be a regular noetherian F_p-algebra. The relative K-groups K_q(A[x]/(x^m),(x)) and the Nil-groups Nil_q(A[x]/(x^m)) were evaluated by the author and Ib Madsen in terms of the big de Rham–Witt groups W_r_A^q of the ring A. In this paper, we evaluate the maps of relative K-groups and Nil-groups induced by the canonical projection f: A[x]/(x^m) -> A[x]/(xⁿ). The result depends strongly on the prime p. It generalizes earlier work by Stienstra on the groups in degrees 2 and 3.

Dynamics, Laplace transform and spectral geometry

Burghelea, D., Haller, S. — 2007-12-17

In this paper, we consider vector fields on a closed manifold whose instantons and closed trajectories can be ‘counted’. Vector fields which admit Lyapunov closed one forms belong to this class. We show that under an additional hypothesis, ‘the exponential growth property’, the counting functions of instantons and closed trajectories have Laplace transforms which can be related to the topology and the geometry of the underlying manifold. The purpose of this paper is to introduce and explore the concept ‘exponential growth property’, and to describe these Laplace transforms.

Heegaard genus and property {tau} for hyperbolic 3-manifolds

Long, D. D., Lubotzky, A., Reid, A. W. — 2007-12-17

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property . As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.

Axioms for higher torsion invariants of smooth bundles

Igusa, K. — 2007-12-17

This paper attempts to explain the relationship between various characteristic classes for smooth manifold bundles which are known as ‘higher torsion’ classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and transfer. We show that higher Franz–Reidemeister torsion and higher Miller–Morita–Mumford classes satisfy these axioms. Conversely, any characteristic class of smooth bundles satisfying the two axioms must be a linear combination of these two examples.

We also show how any higher torsion invariant, that is, any characteristic class satisfying the two axioms, can be computed for a smooth bundle with a fiberwise Morse function with distinct critical values. Finally, we explain the statements of the conjectured formulas relating higher analytic torsion classes, higher Franz–Reidemeister torsion and Dwyer–Weiss–Williams smooth torsion.

Artin groups and the fundamental groups of some moduli spaces

Looijenga, E. — 2007-12-17

We define for every affine Coxeter graph, a certain factor group of the associated Artin group and prove that some of these groups appear as orbifold fundamental groups of moduli spaces. Examples are the moduli space of nonsingular cubic algebraic surfaces and the universal nonhyperelliptic smooth genus three curve. We use this to obtain a simple presentation of the mapping class group of a compact genus three topological surface with connected boundary. This leads to a modification of Wajnryb's presentation of the mapping class groups in the higher genus case that can be understood in algebro-geometric terms.

The homotopy coniveau tower

Levine, M. — 2007-12-17

We examine the ‘homotopy coniveau tower’ for a general cohomology theory on smooth k-schemes and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In addition, we show that the homotopy coniveau tower agrees with Voevodsky's slice tower for S¹-spectra, giving a proof of a connectedness conjecture of Voevodsky. The homotopy coniveau tower construction extends to a tower of functors on the Morel–Voevodsky stable homotopy category, and we identify this P¹-stable homotopy coniveau tower with Voevodsky's slice tower for P¹-spectra. We also show that the zeroth layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general P¹-spectrum the structure of a module over motivic cohomology. This recovers and extends results of Voevodsky on the zeroth layer of the slice filtration, and yields a spectral sequence that is reminiscent of the classical Atiyah–Hirzebruch spectral sequence.