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:
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 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:
Articles were identified using medical, health, occupational and social science databases, including PubMed, OVID, Medline, Proquest, CINAHL plus, CCOHS, SCOPUS, Web of Knowledge, and the following pre-determined criteria were applied: (i) collection of depression measures (as distinct from 'psychological distress') and work status at baseline, (ii) examination and statistical analysis of predictors of work outcomes, (iii) inclusion of cohorts with patients exhibiting symptoms consistent with Acute Coronary Syndrome (ACS), (iv) follow-up of work-specific and depression specific outcomes at minimum 6 months, (v) published in English over the past 15 years. Results from included articles were then evaluated for quality and analysed by comparing effect size.Of the 12 articles meeting criteria, depression significantly predicted reduced likelihood of return to work (RTW) in the majority of studies (n = 7). Further, there was a trend suggesting that increased depression severity was associated with poorer RTW outcomes 6 to 12 months after a cardiac event. Other common significant predictors of RTW were age and patient perceptions of their illness and work performance.Depression is a predictor of work resumption post-MI. As work is a major component of Quality of Life (QOL), this finding has clinical, social, public health and economic implications in the modern era. Targeted depression interventions could facilitate RTW post-MI.Depression is a common and debilitating condition which is often experienced after a heart attack [myocardial infarction (MI)]. It is estimated that approximately 15% of individuals will suffer major depression post-MI, with another 15-20% exhibiting mild to moderate symptoms [1]. Although depression may be transitory, there is evidence to suggest it can precede a cardiac event. For example, more than half of MI patients experience feelings of fatigue and general malaise in the months before infarction [2]. Despite its prevalence, depression often

Abstract:
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 [Trunks and classifying spaces, Applied Categorical Structures, 3 (1995) 321--356]) and the classifying bundle is the first James bundle (defined in "James bundles" math.AT/0301354). 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 \pi_2 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.

Abstract:
We study cubical sets without degeneracies, which we call square sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a square set C has an infinite family of associated square sets J^i(C), i=1,2,..., which we call James complexes. There are mock bundle projections p_i:|J^i(C)|-->|C| (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James--Hopf invariants of Omega(S^2). The algebra of these classes mimics the algebra of the cohomotopy of Omega(S^2) and the reduction to cohomology defines a sequence of natural characteristic classes for a square set. An associated map to BO leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation [M Mahowald, Ring Spectra which are Thom complexes, Duke Math. J. 46 (1979) 549--559] and [B Sanderson, The geometry of Mahowald orientations, SLN 763 (1978) 152--174].