In this note we prove injectivity and relative asphericity for “layered” systems of equations over torsion-free groups, when the exponent matrix is invertible over Z . We also give elementary geometric proofs of results due to Bogley–Pride and Serre that are used in the proof of the main theorem.

The existence of dark matter, inferred from the observed rotation curves of galaxies, is a hypothesis which is widely regarded as problematic. This paper proposes an alternative hypothesis based on the space-time geometry near a rotating body and formulated in terms of the dragging of inertial frames. This hypothesis is true in a certain linear approximation to General Relativity (D Sciama, Mon. Not. Roy. Astron. Soc. 113 (1953) 34--42) and is justified in general by Mach's principle. Dark matter corrects the rotation curve but does not predict the ubiquitous spiral structure of galaxies. The geometric alternative suggested here deals with both problems and allows the construction of a simple model for the dynamics of spiral galaxies which fits observations well.

Welded links were introduced by Fenn, Rimanyi and Rourke in their 1997 Topology paper [2]. In 1999 Kaufmann [7] introduced virtual links. The two concepts are closely connected. Welded links correspond to framed virtual links with one of the two so-called “forbidden moves” allowed. In 2003, in a short elegant paper, “What is a virtual link?”, Kuperberg [8] gave a good geometric description of virtual links: they correspond to links in oriented surfaces cross I and each link has a unique representative with the surface having minimal genus. The purpose of this note is to investigate similar descriptions for welded links. There is no obvious way to extend Kuperberg’s result to welded links, but using ideas of Satoh [11] we can give a satisfactory 4–dimensional interpretation.

In answer to a question from Janos Kollar, we prove that a surjective piecewise-linear map with a topological section admits a nearby homotopic piecewise-linear section.

We propose that a gamma-ray burst is a kinematic effect corresponding to our first entry into the region of space-time illuminated by a continuously emitting object. We illustrate this by an analysis of light rays between time-like geodesics in de Sitter space. In the twoparameter space of pairs of time-like geodesics modulo isometries we find regimes which give light curves similar to observations. We also make remarks about more general models.

This series of papers is devoted to a new approach to the dynamics of galaxies using the precise formulation of Mach’s principle due to Sciama [17], which was discussed in the first paper [11]. In the second paper [12] we applied it to model the rotation curve of galaxies and in this paper we extend the results of [12] to find the full dynamic for orbits in the equatorial plane of a rotating body and use this to model the spiral structure of galaxies.

We outline a model for quasar radiation. The model is based on the simplest black hole accretion model. It allows for significant gravitational redshift, fitting (currently discredited) observations of Arp et al, and provides a natural explanation for the apparently paradoxical phenomena uncovered by Hawkins; it also provides a plausible explanation for the low emissions of Sagittarius A∗ . In order for this model to be plausible, a mechanism for absorbing angular momentum needs to be given. For this we rely on the observation made in [22] that inertial drag allows a black hole to absorb angular momentum.

An observer field in a space-time is a time-like unit vector field. It is natural if the integral curves (field lines) are geodesic and the perpendicular three-plane field is integrable (giving normal space slices). We prove that a natural observer field determines a coherent notion of time: a coordinate that is constant on the perpendicular space slices and whose difference between two space-slices is the proper time along any field line.A natural observer field is flat if the normal space slices are metrically flat. For static spherically symmetric space-times we find a necessary and sufficient condition for possession of a spherically symmetric natural flat observer field. In this case, which includes the Schwarzschild and the Kottler space-times, there is in fact a dual pair of spherically symmetric natural flat observer fields. One of these observer fields is expanding and the other contracting and it is natural to describe the expanding field as the 'escape' field and the dual contracting field as the 'capture' field.Observer fields are useful for understanding redshift and the fields described here are used in a possible explanation of redshift explored in [8].

We use the compression theorem (cf [7; section 6]) to prove results for equivariant con guration spaces analogous to the well-known non-equivariant results of May, Milgram and Segal [5,6,8]. AMS Classi cation 55P91, 55P35, 55P40; 57R91, 55P45, 55P47

This paper is part of a program whose aim is to establish a new paradigm for the universe in which there is no big bang. There are three pieces of primary evidence for the big bang: the distribution of light elements, the cosmic microwave background, and redshift. This paper concentrates on redshift. Alternative explanations for the other pieces of primary evidence are given in other papers in the program.

pinlabel is a labelling package designed for attaching perfectly formatted TEX labels to figures and diagrams in both eps and pdf formats. It is a tool for use by both authors and editors and can be used both for labelling a new diagram and for relabelling an existing diagram. It is the recommended package for (re)labelling diagrams or figures in papers intended for publication by Mathematical Sciences Publishers. The main features of the package are that it uses coordinates read from the diagram in GhostView (or gv) and that labels are placed with automatic and consistent spacing from the object that they are labelling. Many adjustment and positioning options are provided. The end result is a package which is easy and quick to use and which provides accurate and eye-pleasing labelling, completely consistent with the text. For a comparison of pinlabel with other labelling packages which allow one to attach TEX labels, see Section 6. 68-01; 68N01

This paper is about the Compression Theorem: if Mm is embedded in Qq ×R with a normal vector field and if q−m ≥ 1, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q × R. The theorem can be deduced from Gromov’s theorem on directed embeddings [6; 2.4.5 C′ ]. Here we give a direct proof that leads to an explicit description of the finishing embedding. Applications include short new (and constructive) proofs for immersion theory and for the loops–suspension theorem of James et al and a new approach to classifying embeddings of manifolds in codimension one or more, which leads to theoretical solutions. The theorem is relevant to the general problem of simplifying (or specifying) the singularities of a smooth projection up to C–small isotopy and applies to give a theoretical solution in the codimension ≥ 1 case. AMS Classification 57R25, 57R27, 57R40, 57R42, 57R52; 57R20, 57R45, 55P35, 55P40, 55P47

We use Klyachko’s methods [2,3,4,6] to prove that, if a 1–cell and a 2–cell are added to a complex with torsion-free fundamental group, and with the 2–cell attached by an amenable t–shape, then π2 changes by extension of scalars. We also prove that the normal closure of the attaching word contains no words of smaller complexity. AMS Classification 57M20, 57Q05; 20E22, 20F05

Let V be a finite-dimensional real representation of the finite group G and let X be a based G–space with non-degenerate basepoint. There is a wellknown map jV : CV (X) → Ω V S (X) where CV (X) is the space of equivariant configurations in V labelled in X and Ω S (X) is the space of based equivariant maps S → S (X) where S is the 1–point compactification of V and S (X) = S ∧ X with product action. We use the compression theorem (cf [4; section 6]) to prove: (1) If X is G–connected then jV is a weak homotopy equivalence. (2) If V has a trivial summand then jV is a group completion. (3) If V has no trivial summand then CV (X) can be written as X G × A where X is the fixed-point set of X and A is a topological monoid and further jV is equivalent to the natural map X ×A → X×G(A) where G(A) is the group completion of A . Results (1) and (3) are sharper than previously known results [1,6]. AMS Classification numbers Primary: 55P91, 55P35, 55P40 Secondary: 57R91, 55P45, 55P47

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in [10]) and the classifying bundle is the first James bundle (defined in [12]). We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James–Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for π2 of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3–sphere. We also give a cobordism classification of virtual links. AMS Classification 55Q40, 57M25; 57Q45, 57R15, 57R20, 57R40

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in [9]) and the classifying bundle is the first James bundle (defined in [11]). We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James–Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for π2 of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3–sphere. We also give a cobordism classification of virtual links. AMS Classification 55Q40, 57M25; 57Q45, 57R15, 57R20, 57R40

We give a short proof inspired by Carter et al [1] that the 2–twist-spun trefoil is not isotopic to its orientation reverse. The proof uses a computer calculation of the third homology group of the three colour rack. We also give a new proof using the same calculation of the well-known fact that the left and right trefoil knots are not isotopic. AMS Classification 57Q45; 57M25, 57M27