The Betti Numbers of the Free Loop Space of a Connected Sum

Let M₁ and M₂ be two simply connected closed manifolds of thesame dimension. It is proved that (1) if k is a coefficient field such that neither M₁ nor M₂has the same cohomology as a sphere, then the sequence (b_k)_k1of Betti numbers of the free loop space on M₁ #M₂ is unbounded; (2) if, moreover, the cohomology H*(M₁;k) is not generated asalgebra by only one element, then the sequence (b_k)_k1 has anexponential growth. Thanks to theorems of Gromoll and Meyer and of Gromov, thisimplies, in case 1, that there exist infinitely many closedgeodesics on M₁#M₂ for each Riemannian metric, and, in case2, that for a generic metric, the number of closed geodesicsof length t grows exponentially with t.     

Toroidal and Annular Dehn Fillings

Suppose that M is a hyperbolic 3-manifold which admits two Dehnfillings M(r₁) and M(r₂) such that M(r₁) contains an essentialannulus, and M(r₂) contains an essential torus. It is knownthat = (r₁, r₂) 5. We will show that if = 5 then M is theWhitehead sister link exterior, and if = 4 then M is the exteriorof either the Whitehead link or the 2-bridge link associatedto the rational number . There are infinitely many examples with = 3. 1991 Mathematics SubjectClassification: 11D25, 11G05, 14G05.     

Topological lower bounds on the distance between area preserving diffeomorphisms

Area preserving diffeomorphisms of the 2-disk which are Identity near the boundary form a group which can be equipped, using theL ²-norm on its Lie algebra, with a right invariant metric. In this paper we give a lower bound on the distance between diffeomorphisms which is invariant under area preserving changes of coordinates and which improves the lower bound induced by the Calabi invariant. In the case of renormalizable and infinitely renormalizable maps, our estimate can be improved and computed.     

The Extension Group of the Simple Modules Over the First Weyl Algebra

The aim of this paper is to prove that for certain generalizedWeyl algebras A (including the first Weyl algebra A₁ over afield of characteristic zero) and for every simple left (right)A-module M, there are infinitely many non-isomorphic simpleleft (right) A-modules {N_i} such that (respectively ). 1991 Mathematics Subject Classification 18G15, 16G.     

Renormalization in the Hénon family,II: the heteroclinic web

We study highly dissipative Hénon maps

$F_{c,b}: (x,y) \mapsto (c-x^2-by, x)$

with zero entropy. They form a region Π in the parameter plane bounded on the left by the curve W of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in Π, but there exist infinitely many different topological types of such maps (even away from W). We also prove that in the infinitely renormalizable case, the average Jacobian b _F on the attracting Cantor set \({\mathcal{O}}_{F}\) is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside \({\mathcal{O}}_{F}\) if and only if there are no heteroclinic tangencies.

     

On the Representation of Primes by Cubic Polynomials in Two Variables

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002)257–288) we showed that an irreducible integral binarycubic form f(x, y) attains infinitely many prime values, providingthat it has no fixed prime divisor. We now extend this resultby showing that f(m, n) still attains infinitely many primevalues if m and n are restricted by arbitrary congruence conditions,providing that there is still no fixed prime divisor. Two immediate consequences for the solvability of diagonal cubicDiophantine equations are given. 2000 Mathematics Subject Classification11N32 (primary), 11N36, 11R44 (secondary).     

On conformal measures for infinitely renormalizable quadratic polynomials

We study a conformal measure for an infinitely renormalizable quadratic poly- nomial.We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds.The main technique we use in the proof is the three-dimensional puzzle for an infinitely renormalizable quadratic polynomial.     

Local Rigidity of Infinite-Dimensional Teichmuller Spaces

This paper presents a rigidity theorem for infinite-dimensionalBergman spaces of hyperbolic Riemann surfaces, which statesthat the Bergman space A¹(M), for such a Riemann surface M,is isomorphic to the Banach space of summable sequence, l¹.This implies that whenever M and N are Riemann surfaces thatare not analytically finite, and in particular are not necessarilyhomeomorphic, then A¹(M) is isomorphic to A¹(N). It is knownfrom V. Markovic that if there is a linear isometry betweenA¹(M) and A¹(N), for two Riemann surfaces M and N of non-exceptionaltype, then this isometry is induced by a conformal mapping betweenM and N. As a corollary to this rigidity theorem presented here,taking the Banach duals of A¹(M) and l¹ shows that the spaceof holomorphic quadratic differentials on M, Q(M), is isomorphicto the Banach space of bounded sequences, l. As a consequenceof this theorem and the Bers embedding, the Teichmüllerspaces of such Riemann surfaces are locally bi-Lipschitz equivalent.     

Infinitely renormalizable diffeomorphisms of the disk at the boundary of chaos

We show that, among area contracting embeddings of the 2-disk, infinitely renormalizable maps with a bounded geometry either have positive topological entropy or correspond to a cascade of period doubling.

     

Continuity of Actions of Groups and Semigroups on Banach Spaces

It is shown that if a locally compact group acts isometricallyon a Banach space X leaving a closed subspace M invariant, andif the induced actions on M and X/M are strongly continuous,then the action on X is strongly continuous. Since this maybe of interest for one-parameter semigroups, similar resultsare proved for actions of suitable topological semigroups. Othergeneralizations are given for (suitable) non-isometric actions,non-locally compact groups, and non-Banach spaces; corollariesconcerning 1-cocycles and uniformly continuous actions are given.     

A Peak Point Theorem for Uniform Algebras Generated by Smooth Functions on Two-Manifolds

We establish the peak point conjecture for uniform algebrasgenerated by smooth functions on two-manifolds: if A is a uniformalgebra generated by smooth functions on a compact smooth two-manifoldM, such that the maximal ideal space of A is M, and every pointof M is a peak point for A, then A = C(M). We also give an alternativeproof in the case when the algebra A is the uniform closureP(M) of the polynomials on a polynomially convex smooth two-manifoldM lying in a strictly pseudoconvex hypersurface in Cⁿ.     

Interpretable Groups, Stably Embedded Sets, and Vaughtian Pairs

The paper concerns sufficiently saturated structures M overa countable language with a unary predicate P. It is shown thatif P(M)is stably embedded and there are no Vaughtian pairs withrespect to P, then an infinite group is interpretable over M(in an infinitary sense of ‘interpretable’). Also,it is shown that if M is -categorical, f:DP is a 0-definablemap with finite fibres, and P(M) is stably embedded but D isnot, then some infinite group is first-order interpretable overM.     

A trichotomy theorem for o-minimal structures

Let M = M, <, ... be alinearly ordered structure. We defineM to be o-minimal if every definable subset of M is a finiteunion of intervals. Classical examples are ordered divisibleabelian groups and real closed fields. We prove a trichotomytheorem for the structure that an arbitraryo-minimal M can induceon a neighbourhood of any a in M. Roughly said, one of the followingholds:

(i) a is trivial (technical term), or

(ii) a has aconvex neighbourhood on which M induces the structureof anordered vector space, or

(iii) a is contained in an open intervalon which M inducesthe structure of an expansion of a real closedfield.

The proof uses ‘geometric calculus’ whichallows one to recover a differentiable structure by purely geometricmethods. 1991 Mathematics Subject Classification: primary 03C45;secondary 03C52, 12J15, 14P10.     

Homology and derived p-series of groups

We give homological conditions that ensure that a group homomorphisminduces an isomorphism modulo any term of the derived p –series, in analogy to Stallings's 1963 result for the p-lowercentral series. In fact, we prove a stronger theorem that isanalogous to Dwyer's extensions of Stallings’ results.It follows that spaces that are _p-homology equivalent have isomorphicfundamental groups modulo any term of their p-derived series.Various authors have related the ranks of the successive quotientsof the p-lower central series and of the derived p-series ofthe fundamental group of a 3-manifold M to the volume of M,to whether certain subgroups of ₁(M) are free, to whether finiteindex subgroups of ₁(M) map onto non-abelian free groups, andto whether finite covers of M are ‘large’ in variousother senses.     

Bad Fields in Positive Characteristic

If there are infinitely many p-Mersenne prime numbers, thereis no bad field of positive characteristic p. 2000 MathematicsSubject Classification 03C45, 03C60.     

A Note on Osserman Lorentzian Manifolds

Let p be a point of a Lorentzian manifold M. We show that ifM is spacelike Osserman at p, then M has constant sectionalcurvature at p; similarly, if M is timelike Osserman at p, thenM has constant sectional curvature at p. The reverse implicationsare immediate. The timelike case and 4-dimensional spacelikecase were first studied in [3]; we use a different approachto this case. 1991 Mathematics Subject Classification 53B30,53C50.     

The Rationals have an AZ-Enumeration

Let M be an -categorical structure (that is, M is countableand Th(M) is -categorical). A nice enumeration of M is a totalordering of M having order-type and satisfying the following.Whenever a_i, i<, is a sequence of elements from M, thereexist some i<j< and an automorphism of M such that (a_i)= a_j and whenever ba_i, then (b)a_j. Such enumerations were introduced by Ahlbrandt and Ziegler in[1] where they showed that any Grassmannian of an infinite-dimensionalprojective space over a finite field (or of a disintegratedset) admits a nice enumeration; this combinatorial propertyplayed an essential role in their proof that almost stronglyminimal totally categorical structures are quasi-finitely axiomatisable. Recall that if M is -categorical and is a k-tuple of distinctelements from M (with tp() non-algebraic), then the GrassmannianGr(M; ) is defined as follows. The domain of Gr(M; ) is theset of realisations of tp() in M^k, modulo the equivalence relationxEy if x and y are equal as sets. This is a 0-definable subsetof M^eq, and now the relations on Gr(M; ) are by definition preciselythose which are 0-definable in the structure M^eq. (In particular,Gr(M; ) is also -categorical.) Notice that it is by no means clear that if M admits a niceenumeration, then so do Grassmannians of M. However, there isa strengthening of the notion of nice enumeration for whichthis is the case.     

Artinian Rings and the H-Condition

It is proved that a ring R is right Artinian if and only if,for each countably generated right R-module M, there existsa finite subset F of M such that the annihilator of M in R equalsthe annihilator of F in R. 2000 Mathematics Subject Classification16P20.     

On the Number of Carmichael Numbers up to x

It is shown that, for all large x, there are more than x^0.33Carmichael numbers up to x, improving on the ground-breakingwork of Alford, Granville and Pomerance, who were the firstto demonstrate that there are infinitely many such numbers.The same basic construction as that employed by these authorsis used, but a slight modification enables a stronger resulton primes in arithmetic progressions based on a sieve methodto be employed. 2000 Mathematics Subject Classification 11N13(primary), 11N36 (secondary).     

Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices

Starting with an adjoint pair of operators, under suitable abstractversions of standard PDE hypotheses, we consider the Weyl M-functionof extensions of the operators. The extensions are determinedby abstract boundary conditions and we establish results onthe relationship between the M-function as an analytic functionof a spectral parameter and the spectrum of the extension. Wealso give an example where the M-function does not contain thewhole spectral information of the resolvent, and show that theresults can be applied to elliptic PDEs where the M-functioncorresponds to the Dirichlet to Neumann map.