Contraction of Areas vs. Topology of Mappings

We construct homotopically non-trivial maps from S ^m to S ^m?1 with arbitrarily small k-dilation for each k > (m + 1)/2. We prove that homotopically non-trivial maps from S ^m to S ^m?1 cannot have arbitrarily small k-dilation for k ≤ (m + 1)/2.     

Self-maps of the product of two spheres fixing the diagonal

We compute the monoid of essential self-maps of Sn×Sn fixing the diagonal. More generally, we consider products S×S, where S is a suspension. Essential self-maps of S×S demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions.     

On Reeb components of invariant foliations of projectively Anosov flows

We show that if a C² codimension one foliation on a three-dimensional manifold has a Reeb component and is invariant under a projectively Anosov flow, then it must be a Reeb foliation on S²×S¹.     

Cut-disks for level spheres in link and tangle complements

Wu has shown that if a link or a knot L in S³ in thin position has thin spheres, then the thin sphere of lowest width is an essential meridional surface in the link complement. In this paper we show that if we further assume that L⊂S³ is prime, then the thin sphere of lowest width also does not have any vertical cut-disks. We also prove the result for a specific kind of tangles in S²×[−1,1].     

Sets characterized by missing sums and differences

A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S|>|S−S|. We show that the probability that a uniform random subset of {0,1,…,n} is an MSTD set approaches some limit ρ>4.28×¹⁰−4. This improves the previous result of Martin and O?Bryant that there is a lower limit of at least 2×¹⁰−7. Monte Carlo experiments suggest that ρ≈4.5×¹⁰−4. We present a deterministic algorithm that can compute ρ up to arbitrary precision. We also describe the structure of a random MSTD set S⊆{0,1,…,n}. We formalize the intuition that fringe elements are most significant, while middle elements are nearly unrestricted. For instance, the probability that any “middle” element is in S approaches 1/2 as n→∞, confirming a conjecture of Miller, Orosz, and Scheinerman. In general, our results work for any specification on the number of missing sums and the number of missing differences of S, with MSTD sets being a special case.     

On three-manifolds dominated by circle bundles

We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of S ² × S ¹. This characterization can also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. We also determine which three-manifolds are dominated by non-trivial circle bundles, and which three-manifold groups are presentable by products.     

The volume flux group and nonpositive curvature

We show that every closed nonpositively curved manifold with non-trivial volume flux group has zero minimal volume, and admits a finite covering with circle actions whose orbits are homologically essential. This proves a conjecture of Kedra–Kotschick–Morita for this class of manifolds.      

On central difference sets in certain non-abelian 2-groups

In this note, we define the class of finite groups of Suzuki type, which are non-abelian groups of exponent 4 and class 2 with special properties. A group G of Suzuki type with |G|=2^2s always possesses a non-trivial difference set. We show that if s is odd, G possesses a central difference set, whereas if s is even, G has no non-trivial central difference set.     

Algebraic and geometric intersection numbers for free groups

We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of S²×S¹.     

Special manifolds and shape fibrator properties

Our main interest in this paper is further investigation of the concept of (PL) fibrators (introduced by Daverman [R.J. Daverman, PL maps with manifold fibers, J. London Math. Soc. (2) 45 (1992) 180-192]), in a slightly different PL setting. Namely, we are interested in manifolds that can detect approximate fibrations in the new setting. The main results state that every orientable, special (a new class of manifolds that we introduce) PL n-manifold with non-trivial first homology group is a fibrator in the new category, if it is a codimension-2 fibrator (Theorem 8.2) or has a non-cyclic fundamental group (Theorem 8.4). We show that all closed, orientable surface S with χ(S)<0 are fibrators in the new category.     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

Homotopy decomposition of a group of symplectomorphisms of S×S

We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of S²×S² when the ratio of the area of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the pushout (or amalgam) of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.     

Maskit combinations of Poincaré-Einstein metrics

We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics, and also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join. This construction is also extended to spaces with a finite number of interior conic singularities, and as a result we show that any 3-manifold which is a finite connected sum of quotients of S³ and S²×S¹ bounds such a space (with conic singularities); putatively, any 3-manifold admitting a metric of positive scalar curvature is of this form.     

Approximating L 2-Invariants and Homology Growth

In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and torsion invariants. We will show that all these vanish in the limit if the CW-complex under consideration fibers in a specific way. In particular we will show that all these vanish in the limit if one considers an aspherical closed manifold which admits a non-trivial S ¹-action or whose fundamental group contains an infinite normal elementary amenable subgroup. By considering classifying spaces we also get results for groups.     

The topological types of hypersurface simple K3 singularities

We give a result that relates the diffeomorphism type of the link of a non-degenerate semi-quasi-homogeneous hypersurface simple K3 singularity with the singularities of the normal K  3 surface that appears as the exceptional divisor of the resolution of the singularity. As a result, we show that the links are diffeomorphic to the connected sum of copies of S²×S³$S^2\times S^3$. Moreover, we also show that the topological types of hypersurface simple K3 singularities defined by non-degenerate semi-quasi-homogeneous polynomials are all different.     

Invariant subspaces for operator semigroups with commutators of rank at most one

Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of ST−TS is at most 1 for all {S,T}⊂S. We prove that S has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that S is triangularizable if it consists of polynomially compact operators. This generalizes results from [H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997) 443-456] and [G. Cigler, R. Drnovšek, D. Kokol-Bukovšek, T. Laffey, M. Omladi?, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998) 452-465].     

Some remarks on Hilbertian fields (An appendix to the paper “Galois averages” by R. Massy)

The paper gives proofs of some results just claimed in [R. Massy, Galois averages, J. Number Theory 113 (2005) 244-275]. For instance, it is proved that for a finite non-trivial separable extension M/F, M≠F, of Hilbertian fields finitely generated over their prime field, the quotient group M^×/F^×, for the corresponding multiplicative groups of non-zero elements, cannot be a torsion group of finite exponent.     

Some results on b-AM-compact operators

We show that for positive operator B : E → E on Banach lattices, if there exists a positive operator S : E → E such that:1.SB ≤ BS;2.S is quasinilpotent at some x₀ > 0; 3.S dominates a non-zero b-AM-compact operator, then B has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM- compact-friendly operator which is quasinilpotent at some x₀ > 0 has a non-trivial closed invariant ideal.     

Two-body threshold spectral analysis, the critical case

We study in dimension d?2 low-energy spectral and scattering asymptotics for two-body d-dimensional Schrödinger operators with a radially symmetric potential falling off like −γr−2, γ>0. We consider angular momentum sectors, labelled by l=0,1,…, for which γ>²(l+d/2−1). In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.     

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

We study the Gevrey solvability of a class of complex vector fields, defined on Ω?=(−?,?)×S¹, given by L=∂/∂t+(a(x)+ib(x))∂/∂x, b?0, near the characteristic set Σ={0}×S¹. We show that the interplay between the order of vanishing of the functions a and b at x=0 plays a role in the Gevrey solvability.