Obstruction Theory Research Articles

PGSE-NMR studies of solvent diffusion in poly(N-isopropylacrylamide) colloidal microgels

The solvent self-diffusion coefficient has been studied in thermoshrinking poly(N-isopropyl acrylamide) microgel dispersions by the pulsed-gradient spin-echo PGSE-NMR technique, as a function of temperature and mass fraction. After suitable corrections for the temperature, the H2O/D2O ratio and the relative volume fractions, all the self-diffusion data obtained over a temperature range of approximately 40 °C and mass fraction (2–12 % wt/wt) could be superimposed with the volume fraction as the universal factor. The observed reduction in the solvent self-diffusion coefficient with volume fraction was greater than that predicted by simple obstruction theory. After correction for-, and the subsequent removal of the obstruction effect, the diffusion of the solvent through the core of the particle is elucidated. As found for other polymer-solvent systems, there were no specific binding effects. The diffusion of the solvent in these dispersions over such temperature and mass fraction ranges could be rationalised assuming a constant solvent self-diffusion coefficient in the core of the particles.

Higher Homotopy Operations and The Realizability of Homotopy Groups

We describe an obstruction theory for the realization of a Π-algebra—that is, a graded group G* with a prescribed action of the primary homotopy operations—as the homotopy groups of some space. The obstructions consist of higher homotopy operations, for which we provide an explicit definition in terms of certain sequences of polyhedra. There is a similar theory for realizing morphisms between Π-algebras, and thus, in particular, for distinguishing different realizations of a fixed Π-algebra. As an application we show that, for all primes p, the Π-algebra π*Sr⊗Z/p cannot be realized

Obliteration of the distal bile duct in the development of congenital choledochal cyst

The authors report the first case of antenatally diagnosed choledochal cyst having evidence of early fetal bile drainage and complete obliteration of the distal bile duct. This disputes the traditional hypothesis of reflux of pancreatic enzymes and supports the recent theory of primary obstruction as the etiologic cause of choledochal cyst. Coexisting congenital hypothyroidism and pulmonary stenosis had not been reported previously.

Laryngocele Associated with Squamous Carcinoma in a 20-Year-Old Nonsmoker

A laryngocele associated with squamous cell carcinoma of the larynx is reported in a 20-year-old man without a history of smoking. This is the youngest reported case of carcinoma associated with laryngocele in the literature. This association is supported by the patient's young age and his nonsmoking status. However, this case does not appear to support the conclusion that laryngoceles predispose to the development of squamous cell carcinoma. The clinical history supports the tumor obstruction theory, whereby a laryngocele developed from an obstructing carcinoma of the ventricle.

Strong shape of uniform spaces

A strong shape category for finitistic uniform spaces is constructed and it is shown, that certain nice properties known from strong shape theory of compact Hausdorff spaces carry over to this setting. These properties include a characterization of the new category as localization of the homotopy category, the product property and obstruction theory.

Autosomal dominant polycystic kidney disease (ADPKD)--mechanisms of cyst formation and renal failure.

None of the hypotheses proposed so far to explain cyst formation in autosomal dominant polycystic kidney disease (ADPKD) is entirely satisfactory, e.g. the theory of tubular obstruction by intraluminal polyps or dilatation of nephron segments as a consequence of abnormal compliance of the basement membrane. Recent in vitro studies show that (i) synthesis of basement membrane material is abnormal and that (ii) the direction of transepithelial resorptive flux into a secretory mode is reversed as a consequence of faulty insertion of Na, K-ATP'ase into the luminal membrane. It remains unclear why cystic transformation of a few percent of nephrons should cause endstage renal failure. Our clinical and experimental studies do not provide evidence to support some hypotheses proposed in the past, i.e. that renal parenchyma is compressed by expanding cysts and that glomeruli are overperfused. Our histological studies show that progression to endstage renal failure is associated with (i) progressive arteriolar lesions (out of proportion to the vascular lesions seen in extrarenal vascular beds; and (ii) progressive interstitial fibrosis. It appears that fibroblasts in ADPKD are particularly sensitive to platelet derived growth factor (PDGF) which is secreted by epithelial cells of the cyst wall in a paracrine fashion. In contrast to previous opinion, which was presumably skewed by ascertainment bias, it appears that not all, and perhaps not even a majority, of ADPKD patients progress to endstage renal failure. Factors related to progression are gender, family history and hypertension. Both abnormal sodium excretion and inappropriate renin secretion play a role in the genesis of hypertension. Elevated blood pressure, albeit within the normotensive range, is demonstrable even in prepubertal children. The involvement of renin in renal vasoconstriction of normotensive ADPKD patients suggests a particular role of ACE inhibitors in the management of these patients.

Filtered model of a fibration and rational obstruction theory

For a Serre fibration a filtered model is defined and obstructions to the homotopy equivalence of two fibrations (maps), to the section problem and to the extension problem in rational homotopy theory are given.

Generalized Toda Brackets and Equivariant Moore Spectra

In this paper we develop a general theory of obstructions to the existence of equivariant Moore spectra. The obstructions we obtain coincide with higher order Toda brackets as defined by Spanier. We then apply the theory to show the existence of equivariant Moore spectra in various special cases.

Centric Maps and Realization of Diagrams in the Homotopy Category

We study the problem of taking a diagram in the homotopy category and realizing it as a diagram in the category of spaces. Under some assumptions there is a relatively simple obstruction theory for doing this; these assumptions hold in two cases involving homotopy decomposition diagrams for the classifying spaces of compact Lie groups.

Generalized Toda brackets and equivariant Moore spectra

In this paper we develop a general theory of obstructions to the existence of equivariant Moore spectra. The obstructions we obtain coincide with higher order Toda brackets as defined by Spanier. We then apply the theory to show the existence of equivariant Moore spectra in various special cases.

Centric maps and realization of diagrams in the homotopy category

We study the problem of taking a diagram in the homotopy category and realizing it as a diagram in the category of spaces. Under some assumptions there is a relatively simple obstruction theory for doing this; these assumptions hold in two cases involving homotopy decomposition diagrams for the classifying spaces of compact Lie groups.

Maps between iterated loop spaces

An obstruction theory is given which facilities the computation of Dyer-Lashof operations in the homology of iterated loop spaces. This theory investigates the behavior of these operations with respect to maps which are not endowed with enough structure to provide naturality. Examples are given which demonstrate how this technique can be used to compute Dyer-Lashof operations.

Geometric decompositions, algebraic models and rigidity theorems

We explore the (co)formality and (co)spherical generation of a space and the relationship between them. As a consequence of general rigidity results we obtain a linear obstruction theory for these properties. We also show that the skeletal versions of these properties are in fact characterized by this linearization.

ORIENTABILITY OF BUNDLES: OBSTRUCTION THEORY AND APPLICATIONS TOK-THEORY

An obstruction theory is constructed for orientability of vector, piecewise-linear, and topological Rn-bundles and homotopy sphere bundles (spherical fibrations) in generalized cohomology theories. The results are applied to study the orientability of bundles in complex K-theory. In particular, it turns out that the problem of K-orientability for each of the four classes of bundles mentioned above is to be solved differently. Bibliography: 20 titles.

Periodic maps on simply connected four-manifolds

IN THIS paper we describe certain invariants of periodic maps on closed, simply connected four-manifolds with the purpose of classifying such maps up to homeomorphism. The manifolds themselves are classified by Freedman [9] in terms of the intersection pairing on second homology and the Kirby--Sicbcnmann invariant. Gcncralizing Freedman’s result to some nonsimply connected four-manifolds Hambleton and Kreck [IO] have recently shown that in the case of a cyclic fundamental group of odd order the manifold is determined up to homeomorphism by the same two invariants togcthcr with the fundamental group (with the intcrscction pairing being understood now as dcfincd on the torsion free part of the second homology group). A similar statement for cvcn order groups fails to bc true. although its failure is less dramatic for algebraic surfaces as follows from the work by the same authors [I I]. From the point of view of the fundamental group acting on the universal cover these results concern the topological classification of fret periodic maps on simply conncctcd four-manifolds. In gcncral, howcvcr. a periodic map is allowed to have fixed points or periodic points of period smaller than the map period. When this happens, the quotient orbifold has singularities. We shall show that in the cast of isolated singularities one can still classify periodic maps by means of some invariants associated with this orbifold, although in the singular case one must also take into account the torsion part of its second homology. In the more rigid situation of a free periodic map no torsion information is needed. This can bc partially accounted for by the fact that many four-manifolds do not support any such maps. On the other hand, by a result of Edmonds [5]. every closed, simply connected fourmanifold supports infinitely many periodic maps with isolated fixed points. The paper is organized in four sections. In Section 2 we discuss the homotopy classification of locally linear, pseudofree actions of finite groups on simply connected fourmanifolds. There we introduce some terminology and prepare the obstruction theory needed for the results of the next two sections. Our main result is the topological classification of semifree periodic maps of odd period. This is the content of Theorem 3.1. A related but somewhat weaker result for periodic maps of even period is contained in Theorem 3.2. As an application of our techniques, we prove in Section 4 that any locally linear, pscudofree action of a finite group on the complex projective plane is topologically conjugate to a linear action. This last statcmcnt about actions on CP’ has also been asserted in [I93 in the context of finite group actions on the Chern manifold. There we concentrated our efforts on actions without fixed points referring the reader for the proof in the semifrce case to the combined results of [7] and [ 121. This point requires now further elaboration for maps of composite period as the authors of [ 121 have meanwhile added to their result a technical hypothesis of

Immersing projective spaces in Euclidean space

SynopsisUsing theKOℝ/2-theoretic obstruction theory developed in [4] and [5], necessary and sufficient conditions are derived for quaternionic projective spaces ℍPkand odd-dimensional complex projective spaces ℂP2k+1, of real dimensionmsay, to immerse in Euclidean space ℝ2m−1in the range l ≦ 14. The results refine those obtained by Davis and Mahowald ([10, 11]) and earlier authors.

A note on the global structure of supermoduli spaces

We recall some deformation theory of Susy-curves and study obstructions to projectedness of supermoduli spaces, both from a general standpoint and by means of the local “coordinate charts” most commonly used in the physical literature. We prove that these give rise to a projected atlas for genusg=2 only.

Replacing homotopy actions by topological actions. II

A homotopy action of a group G G on a space X X is a homomorphism from G G to the group HAUT ( X ) {\operatorname {HAUT}}(X) of homotopy classes of homotopy equivalences of X X . George Cooke developed an obstruction theory to determine if a homotopy action is equivalent up to homotopy to a topological action. The question studied in this paper is: Given a diagram of spaces with homotopy actions of G G and maps between them that are equivariant up to homotopy, when can the diagram be replaced by a homotopy equivalent diagram of G G -spaces and G G -equivariant maps? We find that the obstruction theory of Cooke has a natural extension to this context.

Whitney-sums (fibre-joins) in over space theory and obstruction theory for cohomology with local coefficients

SynopsisThe generalised Whitney sum (fibre-join) and the h-fibre-join can be defined in topM, the category of spaces over M. We note here some general properties of these constructions, and, as a specific example, we consider the relation between them and the extensions to the topM category of the top h-fibre-sequences F∗ΩB→E ∪ CF→B determined by top fibrations F→E→B. As an application we obtain the truncated local coefficient cohomology sequence for a top fibration which is topM principal fibration: this situation applies, for example, to the various stages of the Postnikov decomposition of a non-simply connected space X, and in this case we have M = K1(π1(X)).

Homotopy spectral sequences and obstructions

For a pointed cosimplicial spaceX •, the author and Kan developed a spectral sequence abutting to the homotopy of the total space TotX •. In this paper,X • is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order obstructions belonging to spectral sequence terms. This applies, for example, to the classical homotopy spectral sequence and obstruction theory for an unpointed mapping space, as well as to the corresponding unstable Adams spectral sequence and associated obstruction theory, which are presented here.