Abstract:
This is the third of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, 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 X R. The theorem can be deduced from Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986) 2.4.5 C'] and the first two parts (math.GT/9712235 and math.GT/0003026) gave proofs. Here we are concerned with applications. We give 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. We also consider the general problem of controlling the singularities of a smooth projection up to C^0--small isotopy and give a theoretical solution in the codimension > 0 case.

Abstract:
This is the second of three papers about the Compression Theorem. We give proofs of Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986); 2.4.5 C'] and of the Normal Deformation Theorem [The compression theorem I; 4.7], arxiv:math.GT/9712235.

Abstract:
We use Klyachko's methods [A funny property of sphere and equations over groups, Comm. in Alg. 21 (1993) 2555--2575] (see also Fenn-Rourke, L'Enseignment Math. 42 (1996) 49--74 and math.GR/9810184 and Cohen-Rourke, math.GR/0009101) 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 pi_2 changes by extension of scalars. It then follows using a result of Bogley and Pride, Proc. Edinburgh Math. Soc. 35 (1992) 1--39, that the resulting fundamental group is also torsion free. We also prove that the normal closure of the attaching word contains no words of smaller complexity.

Abstract:
A homology stratification is a filtered space with local homology groups constant on strata. Despite being used by Goresky and MacPherson [Intersection homology theory: II, Inventiones Mathematicae, 71 (1983) 77-129] in their proof of topological invariance of intersection homology, homology stratifications do not appear to have been studied in any detail and their properties remain obscure. Here we use them to present a simplified version of the Goresky-MacPherson proof valid for PL spaces, and we ask a number of questions. The proof uses a new technique, homology general position, which sheds light on the (open) problem of defining generalised intersection homology.

Abstract:
A new classification theorem for links by the authors and Roger Fenn leads to computable link invariants. As an illustration we distinguish the left and right trefoils and recover the result of Carter et al that the 2-twist-spun trefoil is not isotopic to its orientation reverse. We sketch the proof the classification theorem. Full details will appear elsewhere

Abstract:
We study equations over torsion-free groups in terms of their `t-shape' (the occurences of the variable t in the equation). A t-shape is good if any equation with that shape has a solution. It is an outstanding conjecture that all t-shapes are good. In [Klyachko's methods and the solution of equations over torsion-free groups, l'Enseign. Maths. 42 (1996) 49--74] we proved the conjecture for a large class of t-shapes called amenable. In [Tesselations of S^2 and equations over torsion-free groups, Proc. Edinburgh Maths. Soc. 38 (1995) 485--493] Clifford and Goldstein characterised a class of good t-shapes using a transformation on t-shapes called the Magnus derivative. In this note we introduce an inverse transformation called blowing up. Amenability can be defined using blowing up; moreover the connection with differentiation gives a useful characterisation and implies that the class of amenable t-shapes is strictly larger than the class considered by Clifford and Goldstein.

Abstract:
We use the compression theorem (arxiv:math.GT/9712235) cf section 7, to prove results for equivariant configuration spaces analogous to the well-known non-equivariant results of May, Milgram and Segal.

Abstract:
This the first of a set of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, 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 X R. The theorem can be deduced from Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer-Verlag (1986); 2.4.5 C'] and is implicit in the preceeding discussion. Here we give a direct proof that leads to an explicit description of the finishing embedding. In the second paper in the series we give a proof in the spirit of Gromov's proof and in the third part we give applications.

Abstract:
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.

Abstract:
Let $B_n$ denote the classical braid group on $n$ strands and let the {\em mixed braid group} $B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let $B_{m,\infty}=\cup_nB_{m,n}$. We will describe explicit algebraic moves on $B_{m,\infty}$ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented 3--manifold. The moves depend on a fixed link representing the manifold in $S^3$. More precisely, for link complements the moves are: the two familiar moves of the classical Markov equivalence together with {\em `twisted' conjugation} by certain loops $a_i$. This means premultiplication by ${a_i}^{-1}$ and postmultiplication by a `combed' version of $a_i$. For closed 3--manifolds there is an additional set of {\it `combed' band moves} which correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov Theorem using {\it $L$--moves} \cite{LR} (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov Theorem that classifies links in $S^3$ up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of 3--manifolds.