Let K⊂C be a field finitely generated over Q, K(a)⊂C the algebraic closure of K and G(K)=Gal (K(a)/K) its Galois group. For each positive integer m we write K(μm) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime [lscr ] we write K([lscr ]) for the subfield of K(a) obtained by adjoining to K all [lscr ]-power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab)⊂K(a) be the maximal abelian extension of K. The field K(ab) contains K(c); if K=Q then Q(ab)=Q(c) (the Kronecker-Weber theorem). We write χ[lscr ][ratio ]G(K)→Z*[lscr ] for the cyclotomic character defining the Galois action on all [lscr ]-power roots of unity. We write χ[lscr ]= χ[lscr ] mod [lscr ][ratio ]G(K) →Z*[lscr ]→(Z/[lscr ]Z)* for the cyclotomic character defining the Galois action on the [lscr ]th roots of unity. The character χ[lscr ] identifies Gal (K([lscr ])/K) with a subgroup of Z*[lscr ]=Gal (Q([lscr ])/Q). Let μ(Z[lscr ]) be the finite cyclic group μ(Z[lscr ]) of all roots of unity in Z*[lscr ]. Its order is equal to [lscr ]−1 if [lscr ] is odd and 2 if [lscr ]=2. Let Q([lscr ])′ be the subfield of μ(Z[lscr ])-invariants in Q([lscr ]). Clearly, Gal (Q([lscr ])/Q([lscr ])′)=μ(Z[lscr ]) and Gal (Q([lscr ])′/Q)= Z*[lscr ]/μ(Z[lscr ]) is isomorphic to Z[lscr ].

We prove that if q is an integer greater than one, r and s are any positive rationals, then

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is irrational and is not a Liouville number.

The subgroup structure of a direct product of two Fuchsian groups is very complicated; for example, Baumslag and Roseblade [1] have shown that a direct product Fm1×Fm2 of finitely generated free groups contains continuously many distinct isomorphism classes of finitely generated subgroups. The situation is much simpler, however, if attention is restricted to finitely generated normal subgroups of Fm1×Fm2; then on general grounds one cannot expect to get more than a countable infinity, but, in fact, the situation is almost finite; we showed, in [4], that Fm1×Fm2 contains precisely 1+min{m1, m2} orbits of maximal normal subdirect products under the natural action of its automorphism group.

In this paper we study the situation which arises if the free groups Fmi are replaced by fundamental groups of closed surfaces. This question was previously considered by Nigel Carr in the final chapter of his (unpublished) thesis [2]. By appealing to the relative invariant theory of pairs of skew bilinear forms, Carr was able to show that in a direct product Σg1×Σg2 of orientable surface groups of genus [ges ]2 the number of orbits of maximal normal subdirect products is always infinite. Here we refine and extend Carr's approach to study the manner in which the finiteness result of [4] breaks down on passing to more general Fuchsian groups.

The problem of constructing non-diagonal solutions to systems of symmetric diagonal equations has attracted intense investigation for centuries (see [5, 6] for a history of such problems) and remains a topic of current interest (see, for example, [2–4]). In contrast, the problem of bounding the number of such non-diagonal solutions has commanded attention only comparatively recently, the first non-trivial estimates having been obtained around thirty years ago through the sieve methods applied by Hooley [10, 11] and Greaves [7] in their investigations concerning sums of two kth powers. As a further contribution to the problem of establishing the paucity of non-diagonal solutions in certain systems of diagonal diophantine equations, in this paper we bound the number of non-diagonal solutions of a system of simultaneous quadratic and biquadratic equations. Let S(P) denote the number of solutions of the simultaneous diophantine equations

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with 0[les ]xi, yi[les ]P(1[les ]i[les ]3), and let T(P) denote the corresponding number of solutions with (x1, x2, x3) a permutation of (y1, y2, y3). In Section 4 below we establish the upper and lower bounds for S(P)−T(P) contained in the following theorem.

Introduite par Hall [4] en 1979 et étudiée de manière systématique par Hall et Tenenbaum ([6, 7]), la fonction

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est une mesure quadratique de la proximité des diviseurs.

On a trivialement τ(n)[les ]T(n, α)[les ]T(n, 1)=τ(n)2 pour tout α, où, ici et dans la suite, τ(n) désigne le nombre de diviseurs d'un entier générique n: Le statut de l'inégalité de gauche pour presque tout entier n est une question difficile qui a inspiré une conjecture d'Erdős (cf. [1]) assez connue. Convenons de désigner par pp (presque partout) une relation valable sur un ensemble d'entiers de densité unité et posons

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(Cette fonction est mentionnée dans [4], mais sa définition fait l'objet d'une coquille.)

For a non-zero integer n, let d(n) denote the number of positive divisors of n. Let a, b and c be integers with a>0, and set Δ=b2−4ac. If the quadratic polynomial ax2+bx+c is irreducible over the rational numbers Q (that is, if Δ is not the square of an integer), then one has

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as X→∞, for some λ depending on a, b and c (see [7]). In this paper we discuss the way in which λ depends on a, b and c, giving a precise, compact expression in terms of class numbers. This extends previous work for the case a=1, Δ<0 (see [4]).

For the case a=1, b=0, a much better description of the error is given in [2], with the following expression for λ:

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Here ρ is a multiplicative function, defined below, and (p/q) is the Legendre/Jacobi symbol.

We compute the (uni)versal deformation ring of an odd Galois representation ρ: Gal (M/Q) → Gl2(Fp) with an upper triangular image, where M is the maximal abelian pro-p-extension of F∞ unramified outside a finite set of places S, F∞ being a free pro-p-extension of a subextension F of the field K fixed by Ker ρ. We establish a link between the latter (uni)versal deformation ring and the (uni)versal deformation ring of ρ: Gal (KS/Q) → Gl2(Fp), where KS is the maximal pro-p-extension of K unramified outside S. We then give some examples.

In the tangle model for DNA site-specific recombination, one is required to solve simultaneous equations for unknown tangles which are summands of observed DNA knots and links. For 0[les ]i[les ]3, given fixed 4-plats Ki where the set {K1, K2, K3} contains at least two distinct 4-plats, let O and R denote unknown 2-string tangles such that {O, R} are the variables in the system of four tangle equations N(O+iR)=Ki, where N is the numerator construction, and nR denotes the tangle sum of n copies of R. Then there is at most one simultaneous solution {O, R} and this solution must be of the form R an integral tangle and O either a rational tangle or the sum of two rational tangles. In addition, if there exists a solution, then at least one of the 4-plats is chiral. We exhibit an algorithm for solving simultaneous tangle equations of this form.

Let [gscr ] be one of the Lie algebras [sscr ][Iscr ]n(K) or [sscr ][oscr ]2n+1(K) over an algebraically closed field K of characteristic p>0. Suppose in the first case that n∉Zp and in the second case that p≠2. This assumption implies that [gscr ] is simple. In this paper I study certain irreducible representations of [gscr ].

The metric projection mapping πX plays an important role in nonlinear approximation theory. Usually X is a closed subset of a Banach space [ ] and, for each e∈[ ], πX(e) is the set, perhaps empty, of all points in X which are nearest to e. From a classical theorem due to Stečkin [7] it is known that, when [ ] is uniformly convex, the metric projection πX(e) is single valued at each typical point e of [ ] (in the sense of the Baire categories), i.e. at each point e of a residual subset of [ ]. More recently Zamfirescu [8] has proven that, if X is a typical compact set in ℝn (in the sense of Baire categories) and n[ges ]2, then the metric projection πX(e) has cardinality at least 2 at each point e of a dense subset of ℝn. This result has been extended in several directions by Zhivkov [9, 10], who has also considered the case of the metric antiprojection mapping νX (which associates with each e∈[ ] the set νX(e), perhaps empty, of all ∈X which are farthest from e). For this mapping De Blasi [2] has shown that, if [ ] is a real separable Hilbert space with dim[ ]=+∞ and n is an arbitrary natural number not less than 2, then, for a typical compact convex set X⊂[ ], the metric antiprojection νX(e) has cardinality at least n at each point e of a dense subset of [ ]. A systematic discussion of the properties of the maps πX and νX, and additional bibliography, can be found in Singer [5, 6] and Dontchev and Zolezzi [3].

In the present paper we consider some further properties of the metric projection mapping πX, with X a compact set in a real separable Hilbert space [ ]. If dim[ ]=n and 2[ges ]n<+∞, it is proven that for a typical compact set X⊂[ ], the metric projection πX(e) has cardinality exactly n+1 at each point e of a dense subset of [ ], while the set of those points e∈[ ] where πX(e) has cardinality at least n+2 is empty. Furthermore it is shown that, if dim[ ]=+∞, then for a typical compact set X⊂[ ] the metric projection πX(e) has cardinality at least n (for arbitrary n[ges ]2) at each point e of a dense subset of [ ]. Incidentally we obtain a characterization of the dimension of the space [ ] by means of a typical property holding in the space of the compact subsets of [ ].

In this paper we consider reduced curves Γ of fixed degree d in the complex projective plane P2([Copf ]): we study the question of how singular such a curve can be.

The Mullineux map is an involutory bijection on the set of p-regular partitions of any given integer n, where a partition is called p-regular if no part of it is repeated p or more times. Many combinatorial properties of the Mullineux map make it reasonable to view this map as a p-analogue of the transposition map T on the set of all partitions.

Based on the work of Kleshchev [7], it has been shown [2, 5, 14] that the Mullineux map M has the following property.

If λ is a p-regular partition of n and Dλ is the p-modular representation of the symmetric group Sn labelled by λ (see [6]) then

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where sgn is the sign representation of Sn. Thus, from a representation theoretic point of view as well, the Mullineux map is a p-analogue of the transposition map T. The Mullineux map plays a vital role not only in the representation theory of symmetric groups but also in other contexts. The definition of M as given in [9] is quite complicated and to study various questions involving this map it is desirable to have other descriptions. One alternative inductive description of M using the concept of good boxes in a p-regular partition was given by Kleshchev [7] and this was used by Walker to prove a result contained in Theorem 4·5 below; this work was motivated by the investigation of Schur modules. In this paper we study a third description of M based on the operator J on the set of p-regular partitions defined in [13].

This paper relates skein spaces based on the Kauffman bracket and spin structures. A spin structure on an oriented 3-manifold provides an isomorphism between the skein space for parameter A and the skein space for parameter −A.

There is an application to Penrose's binor calculus, which is related to the tensor calculus of representations of SU(2). The perspective developed here is that this tensor calculus is actually a calculus of spinors on the plane and the matrices are determined by a type of spinor transport which generalizes to links in any 3-manifold.

A second application shows that there is a skein space which is the algebra of functions on the set of spin structures for the 3-manifold.

Let G be a simple algebraic group over an algebraically closed field K of characteristic p. If Σ is the root system of G and Uα is the root subgroup of G corresponding to a long root α∈Σ, then 〈Uα, U−α〉 is an image of SL2 and any G-conjugate of this subgroup is called a fundamental subgroup of G. In [LS1], the closed connected semisimple subgroups of G generated by long root elements were determined. Of course, long root elements are unipotent elements of fundamental subgroups. In this paper we consider subgroups of G which are generated by semisimple elements lying in fundamental subgroups.

Our first three results (Theorems 1–3) concern semisimple connected subgroups of G which contain a maximal torus of a fundamental subgroup; we call such a torus a fundamental torus. Notice that for classical groups, the elements in fundamental tori have fixed spaces of small codimension in the natural module; indeed, for the groups SL(V) and Sp(V), the elements of fundamental tori are precisely those semisimple elements with fixed space of codimension 2, while for SO(V), the codimension is 4.

As consequences of these results, we obtain information on subgroups (finite or infinite) of classical groups which are generated by conjugates of a single element of a fundamental torus of order at least 5 (see Theorems 4, 5 and Corollary 6); for example, Theorem 5 determines those finite irreducible subgroups containing such an element which are quasisimple and of Lie type in characteristic p.

We now state our results in detail. Recall from [LS1] that a subsystem subgroup of G is a connected semisimple subgroup which is invariant under a maximal torus of G.

The main result of this paper is to show that if G∗ is a simplicial group of finite length, then HnG∗ also has finite length. Here the length of a simplicial group means the length of the corresponding Moore normalization and HnG∗ is the simplicial abelian group given by [k][map ]HnGk. A similar fact is true if we replace G∗ by a simplicial ring and we take the algebraic K-functors instead of group homology.

The origin of such results goes back to the classical paper of Dold and Puppe (see Hilfsatz 4·23 of [7]), where the following was proved: let

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be a functor between abelian categories of degree d, meaning that the (d+1)st cross-effect functor in the sense of Eilenberg and MacLane [9] vanishes. Then l(TX∗)[les ]dl(X∗) for any simplicial object X∗ in A. Here l(X∗) denotes the length of X∗. Actually, this property characterizes the degree of functors in the abelian case (see Lemma 3·6).

One can modify the notion of the cross-effects in the nonabelian framework (see [2]) and define the notion of degree of functors. One can also use the above inequality to define the simplicial degree in the nonabelian set-up. However in this way we get generally different invariants and the aim of this paper is to establish the relationship between the different notions. We show that many classical functors arising in homological algebra and K-theory have finite simplicial degree. Our Conjecture 4·7 claims that this should be always the case.

We remark that if the Moore normalization of a simplicial group G∗ is zero in dimensions >k then πiG∗=0 for i>k as well. Therefore, for a functor T of simplicial degree d, one has

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Our work can therefore be considered as a new general method proving vanishing results. Here is a sample application of the main result.

We give a construction of Gusarov's groups [Gscr ]n of knots based on pure braid commutators, and show that any element of [Gscr ]n is represented by an infinite number of prime alternating knots of braid index less than or equal to n+1. We also study [Vscr ]n, the torsion-free part of [Gscr ]n, which is the group of equivalence classes of knots which cannot be distinguished by any rational Vassiliev invariant of order less than or equal to n. Generalizing the Gusarov–Ohyama definition of n-triviality, we give a characterization of the elements of the nth group of the lower central series of an arbitrary group.

In the early seventies, Steenrod posed the question: which polynomial algebras over the Steenrod algebra appear as the cohomology ring of a topological space? For odd primes, work of Adams and Wilkerson and Dwyer, Miller and Wilkerson showed that all such algebras are given as the mod-p reduction of the invariants of a pseudo reflection group acting on a polynomial algebra over the p-adic integers. We show that this necessary condition is also sufficient for finding a realization of such a polynomial algebra. We also show that, up to completion, every space with polynomial mod-p cohomology is equivalent to the product of the classifying space of a connected compact Lie group and some spaces determined by irreducible p-adic rational pseudo reflection groups.

Let F be a free group. We explain the classification of finitely presented subgroups of F×F in geometric terms. The classification emerges as a special case of results concerning the structure of 2-complexes which are built out of squares and have the property that the link of each vertex has no reduced circuits whose length is odd or less than four. We obtain these results using tower arguments and elements of the theory of non-positively curved spaces.

It is a well known fact that every group G has a presentation of the form G=F/R, where F is a free group and R the kernel of the natural epimorphism from F onto G. Driven by the desire to obtain a similar presentation of the group of automorphisms Aut (G), we consider the subgroup Stab (R) ⊆ Aut (F) of those automorphisms of F that stabilize R and ask whether the natural homomorphism Stab (R) → Aut (G) is onto; if it is, we can try to determine its kernel.

Both parts of this task are usually quite hard. The former part received considerable attention in the past, whereas the more difficult part (determining the kernel) seemed unapproachable. Here we approach this problem for a class of one-relator groups with a special kind of small cancellation condition. Then, we address a somewhat easier case of 2-generator (not necessarily one-relator) groups and determine the kernel of the above-mentioned homomorphism for a rather general class of those groups.

Let V⊂S3 be a solid, knotted torus. Through the work of Birman and Menasco [2], the observation has been made that a satellite link L=C[midast ]P⊂V (with companion C, pattern P and essential torus T=∂V) falls into one of two broad categories: reverse string and non-reverse string. These categories are borne of the three embedding types identified in [2] and are distinguished by the existence of, or lack of, a meridional disc D⊂V with an orientation, whose point-intersections with the oriented satellite C[midast ]P are all similarly oriented. If no such D exists, then L is said to be a reverse string satellite.

If C[midast ]P is a non-reverse string satellite, it is known that the braid index b(C[midast ]P) is dependent only on b(C) and certain specified properties of the pattern and not on the choice f∈Z of framing parameter. In [2] it is conjectured that, if C[midast ]fP is a reverse string satellite (in which the framing parameter is f), the braid index b(C[midast ]fP) depends on the arc index α(C), properties of the pattern and also the framing. We study this dependence further via established upper and lower bounds for braid index. The upper bound comes from explicit closed braid diagrams of L, for which constructions are shown. The lower bound comes from the Homfly polynomial, via the Morton–Franks–Williams inequality [5, 8]. We will prove the following theorem.