Let k be an algebraically closed field of arbitrary characteristic. Lines in ℙ3 are parametrized by the Grassmannian G(2, 4), which is isomorphic to a smooth quadric in ℙ5. We can consider the configuration space Xn = G(2, 4)n / PGL4(k) parametrizing ordered n-tuples of lines in ℙ3 up to projective equivalence. dim PGL4(k) = 15 and for n [ges ] 5, the stabilizer of a general n-tuple of lines is trivial, so for n [ges ] 5, Xn has the expected dimension 4n − 15.

The question of rationality of Xn was posed by Dolgachev. The space Xn is clearly unirational, since there is a dominant rational map to it from the rational variety G(2, 4)n. The following results are known in characteristic 0: it is a special case of a theorem by Dolgachev and Boden [1] for configuration spaces in greater generality that if Xn is rational for some n [ges ] 5 then so is XN for any N [ges ] n. They also proved that the configuration space of lines in ℙm is rational if m is odd and recently Zaitsev [2] proved this for all m.

Our proof uses different methods and it also has the advantage that it is valid in any characteristic. The main result is the following:

We classify finite Moufang loops which are centrally nilpotent of class 2 in terms of certain cubic forms, concentrating on small Frattini Moufang loops, or SFMLs, which are Moufang loops L with a central subgroup Z of order p such that L/Z is an elementary abelian p-group. (For example, SFM 2-loops are precisely the class of code loops, in the sense of Griess.)

More specifically, we first show that the nuclearly-derived subloop (normal associator subloop) of a Moufang loop of class 2 has exponent dividing 6. It follows that the subloop of elements of p-power order is associative for p > 3. Next, we show that if L is an SFML, then L/Z has the structure of a vector space with a symplectic cubic form. We then show that every symplectic cubic form is realized by some SFML and that two SFMLs are isomorphic in a manner preserving the central subgroup Z if and only if their symplectic cubic spaces are isomorphic up to scalar multiple. Consequently, we also obtain an explicit characterization of isotopy in SFM 3-loops. Finally, we extend many of our results to all finite Moufang loops of class 2.

Code loops are certain Moufang loop extensions of doubly even binary codes which are useful in finite group theory (e.g. Conway's construction of the Monster). We give several methods for explicitly constructing code loops as centrally twisted products. More specifically, after establishing some preliminary examples, we show how to use decompositions of codes to build code loops out of more familiar pieces, such as abelian groups, extraspecial groups, or the octonion loop. In particular, we use Turyn's construction of the Golay code to give a simple explicit construction of the Parker loop, one which may have applications to the study of the Monster.

In this paper we extend the concept of the group of covering automorphisms associated to a universal covering space ϕ: U → X (where X is a connected topological manifold), to the case of left (or right) minimal approximations. In the case of torsion-free coverings of abelian groups we exhibit a topology on these groups which makes them into topological groups and we give necessary and sufficient conditions for these groups to be compact. Finally we prove that when these groups are compact they are pronilpotent (Theorem 5·3). We also characterize when these groups are torsion-free (Proposition 5·4).

A two-generator group 〈f, g〉 of Möbius maps is determined, up to conjugacy, by the numbers β = tr2(f) − 4, β′ = tr2(g) − 4 and γ = tr (fgf−1g−1) − 2, provided that γ ≠ 0. We study the subset D of C3 of those (β, β′, γ) which arise from discrete groups. In particular, we identify precisely D[setmn ]D.

In this paper we investigate the relationship between genus and localization of finitely generated (torsion-free abelian)-by-finite groups and genus and localization in integral representation theory. In particular when such a group has finite commutator subgroup we can determine the genus exactly.

Throughout this paper, let M be a finitely generated module over a Noether local ring (A, [mfr ]) with dim M = d. Let x = (x1, …, xd) be a system of parameters of M and n = (n1, …, nd) ∈ ℕd a d-tuple of positive integers. This paper is concerned with the following two points of view.

First, it is well-known that, the difference between lengths and multiplicities

formula here

considered as a function in n, gives a lot of information on the structure of M. This function in general is not a polynomial in n for all n1, …, nd large enough (n [Gt ] 0 for short). But, it was shown in [C3] that the least degree of all polynomials in n bounding above IM (n, x) is independent of the choice of x. This numerical invariant is denoted by p(M) and called the polynomial type of the module M. By [C2] and [C3] this polynomial type does not change by the [mfr ]-adic completion Mˆ of M and p(M) is just equal to the dimension of the non-Cohen–Macaulay locus of Mˆ (see [C2, C3, C4, CM] for more details). Therefore a module M is Cohen–Macaulay or generalized Cohen–Macaulay if and only if p(M) = − ∞ or p(M) [les ] 0, respectively, where we set by − ∞ the degree of the zero-polynomial. However, one knows little about the structure of M when p(M) > 0.

Second, following Sharp and Hamieh ([SH]), we consider the difference

formula here

as a function in n, where

formula here

is the cyclic submodule of the module of generalized fractions U(M)−d−1d+1M defined in [SZ1].

It is a classical result of Postnikov [15] that homotopy types X with at most two non-trivial homotopy groups πmX = A and πnX = B, 2 [les ] m < n, are classified by the k-invariant

formula here

Here the cohomology group of the Eilenberg–MacLane space K(A, m) was computed by Eilenberg–MacLane [11] and Cartan [5]. Let p be a prime and let ℤp ⊂ ℚ be the smallest subring of ℚ containing 1/q for all primes q with q ≠ p. We consider finitely generated ℤp-modules A and B and the stable range n < 2m − 1. Hence X is a p-local space with at most two non-trivial homotopy groups in a stable range. Then the homotopy type of X admits a product decomposition

formula here

where all Xi with 1 [les ] i [les ] j are indecomposable and this decomposition is unique up to permutation. We classify in this paper the indecomposable factors in (2) by the following result.

We establish a connection between the η invariant of Atiyah, Patodi and Singer ([1, 2]) and the condition that a knot K ⊂ S3 be slice. We produce a new family of metabelian obstructions to slicing K such as those first developed by Casson and Gordon in [4] in the mid 1970s. Surgery is used to turn the knot complement S3 − K into a closed manifold M and, for given unitary representations of π1(M), η can be defined. Levine has recently shown in [11] that η acts as an homology cobordism invariant for a certain subvariety of the representation space of π1(N), where N is zero-framed surgery on a knot concordance. We demonstrate a large family of such representations, show they are extensions of similar representations on the boundary of N and prove that for slice knots, the value of η defined by these representations must vanish.

The paper is organized as follows; Section 1 consists of background material on η and Levine's work on how it is used as a concordance invariant [11]. Section 2 deals with unitary representations of π1(M) and is broken into two parts. In 2·1, homomorphisms from π1(M) to a metabelian group Γ are developed using the Blanchfield pairing. Unitary representations of Γ are then considered in 2·2. Conditions ensuring that such two stage representations of π1(M) allow η to be used as an invariant are developed in Section 3 and [Pscr ]k, the family of such representations, is defined. Section 4 contains the main result of the paper, Theorem 4·3. Lastly, in Section 5, we demonstrate the construction of representations in [Pscr ]k.

Suppose [Cscr ] and [Cscr ]′ are two sets of simple closed curves on a hyperbolic surface F. We give necessary and sufficient conditions for the existence of a pseudo-Anosov map g such that g([Cscr ]) ≅ [Cscr ]′.

The symmetric imprimitivity theorem provides a Morita equivalence between two crossed products of induced C*-algebras and includes as special cases many other important Morita equivalences such as Green's imprimitivity theorem. We show that the symmetric imprimitivity theorem is compatible with various inflated actions and coactions on the crossed products.

The Pisier algebra [Pscr ] consists of those continuous functions f on the unit circle [ ] for which

formula here

is continuous on [ ] almost surely, that is, for almost every choice of a sequence of signs (±1, ±1, …). In this paper, we prove that spectral synthesis holds in [Pscr ]. Moreover, we show that certain closed ideals in [Pscr ] with infinite hull have bounded approximate identities and give an example of a closed ideal in [Pscr ] which does not have a bounded approximate identity.

Let f: C → C denote a transcendental entire function and let m(r, f) equal min {[mid ]f(z)[mid ][ratio ][mid ]z[mid ] = r}. In this note, we mainly prove that if

formula here

then the set of normality N(f) of f has no unbounded preperiodic and periodic components. In particular, f has no Baker domains. We also discuss non-existence of unbounded components of normality under other conditions.

Equilibrium states in galactic dynamics can be described as stationary solutions of the Vlasov–Poisson system, which is the non-relativistic case, or of the Vlasov–Einstein system, which is the relativistic case. To obtain spherically symmetric stationary solutions the distribution function of the particles (stars) on phase space is taken to be a function Φ(E, L) of the particle energy and angular momentum. We give a new condition on Φ which guarantees that the resulting steady state has finite mass and compact support both for the non-relativistic and the relativistic case. The condition is local in the sense that only the asymptotic behaviour of Φ for E → E0 needs to be prescribed, where E0 is a cut-off energy above which no particles exist.

(Volume 125 (1999), 433–440)

On page 438 of this article Figure 1 was incorrectly printed. The correct version appears below.

Fig. 1.

The Editors and Publishers apologise for any confusion this oversight may have caused.