Obstruction Theory Research Articles

Equivariant Surgery Theory for Homology Equivalences under the Gap Condition

In the present paper, we discuss an obstruction theory to modify equivariant framed maps on even-dimensional compact smooth manifolds to homology equivalences by equivariant surgery. In 1974, Cappell–Shaneson already developed such obstruction theory in the nonequivariant setting. Our definition of the surgeryobstruction group presents a new aspect of Cappell–Shaneson’s group in the nonequivariant setting and enables us to define directly the surgery obstructions of certain framed maps that are not necessarily connected up to the middle dimension. Using our framework defining the equivariant surgery obstruction, we prove a basic conjecture related to geometric connected sums and algebraic sums of surgery obstructions.

On realizing diagrams of Π–algebras

Given a diagram of Pi-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Pi-algebras. This extends a program begun in [J. Pure Appl. Alg. 103 (1995) 167-188] and [Topology 43 (2004) 857-892] to study the realization of a single Pi-algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations.

Obstruction Theory and Coincidences in Positive Codimension

Let f, g : X → Y be two maps between closed manifolds with dim X ≥ dim Y = n ≥ 3. We study the primary obstruction on(f, g) to deforming f and g to be coincidence free on the n–th skeleton of X. We give examples for which obstructions to deforming f and g to be coincidence free are detected by on(f, g).

Bredon-style homology, cohomology and Riemann–Roch for algebraic stacks

One of the main obstacles for proving Riemann–Roch for algebraic stacks is the lack of cohomology and homology theories that are closer to the K-theory and G-theory of algebraic stacks than the traditional cohomology and homology theories for algebraic stacks. In this paper we study in detail a family of cohomology and homology theories which we call Bredon-style theories that are of this type and in the spirit of the classical Bredon cohomology and homology theories defined for the actions of compact topological groups on topological spaces. We establish Riemann–Roch theorems in this setting: it is shown elsewhere that such Riemann–Roch theorems provide a powerful tool for deriving formulae involving virtual fundamental classes associated to dg-stacks, for example, moduli stacks of stable curves provided with a virtual structure sheaf associated to a perfect obstruction theory. We conclude the present paper with a brief application of this nature.

Interpolation categories for homology theories

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions, we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolution model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams–Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As an application, we establish an isomorphism between certain E ( n ) -local Picard groups and some Ext-groups.

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}$ that lifts to ℤ/p 7 but not ℤ/p 8. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mnev’s universality theorem.

Equipartition of sphere measures by hyperplanes

Measure partition problems are classical problems of geometric combinatorics ([1], [2], [3], [4]) whose solutions often use tools from the equivariant algebraic topology. The potential of the computational obstruction theory approach is partially demonstrated here. In this paper we reprove a result of V.V. Makeev [9] about a 6-equipartition of a measure on S2 by three planes. The advantage of our approach is that it can be applied on other more complicated questions of the similar nature.

On the $\Gamma$-cohomology of rings of numerical polynomials

We investigate \Gamma -cohomology of some commutative cooperation algebras E_*E associated with certain periodic cohomology theories. For KU and E(1) , the Adams summand at a prime p , and for KO we show that \Gamma -cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E_\infty structures. As a consequence we obtain an E_\infty structure for the connective Adams summand. For the Johnson–Wilson spectrum E(n) with n\geq1 we establish the existence of a unique E_\infty structure for its I_n -adic completion.

Obstructive theory of disturbed external breathing. State of the art, perspectives of the development

The development of obstructive theory and methods of pulmonary ventilation function studying with demonstrating of main dialectical laws: the development from simple to complicated, from the test using candle blowing out in the beginning of the XIX century to the pletismography of the body and breathing mechanics in the middle of XX century. The spiral finishes to the beginning of the XXI century by using confirmed methods to assess forced output assessment for the 1-st second and peakflowmetry. The negation of negation law is seen here. The development of obstructive theory took place in the framework of Donders’ paradigm (1853). Fundamental controversies of the paradigm have been analysed, the theory of pulmonary mechanical activity is validated which is anticipated to be a new paradigm and may and must provide the development of breathing clinical physiology to more complex breathing mechanics and to complete a new spiral by the development of delta-peakflowmetry designing.

Topological robotics in lens spaces

Motivated by the work of Farber, Tabachnikov and Yuzvinsky on the motion planning problem for projective spaces, we give an estimate for the topological complexity (TC) of lens spaces in terms of certain generalized “skew” maps between spheres. This last concept turns out to be closely related to that for a generalized axial map developed by Astey, Davis and the author to characterize the smallest Euclidean dimension where (2-torsion) lens spaces can be immersed. As a result, this suggests an alternative simpler “TC-approach” to the classical immersion problem for real projective spaces, whose initial stages we settle by means of techniques in obstruction theory.

Obstruction Theory for Objects in Abelian and Derived Categories#

ABSTRACT In this article, we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case we also obtain the corresponding obstruction theory for lifting of objects in terms of Yoneda Ext-groups. In an appendix we prove the existence of miniversal derived deformations of complexes.

Obstruction theory and coincidences of maps between nilmanifolds

Let f, g : M → N be two maps between two compact nilmanifolds with dim M ≧ dim N = n. In this paper, we show that either the Nielsen coincidence number N(f, g) = 0 or N(f, g) = R(f, g) where R(f, g) denotes the Reidemeister number of f and g. Furthermore, we show that if N(f, g) > 0 then the primary obstruction o n (f, g) to deforming f and g to be coincidence free on the n-th skeleton of M is non-trivial.

Obstruction Theory for the Approximation and the Deformation Problems for Sobolev Mappings

We develop the obstruction theory for Sobolev mappings both for the approximation and the deformation problems. We prove the existence of the ‘primary obstruction class’ for the deformation problem and classify the local and the global obstructions for the approximation problem. We also give precise descriptions of these obstructions when the domain is a simply connected 4-manifold.

Hochschild cohomology of abelian categories and ringed spaces

This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in complete generality that Hochschild cohomology satisfies a Mayer–Vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf.

Equivariant Extensions of Categorical Groups

If Γ is a group, then the category of Γ-graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from Γ. In this paper we use this equivalence and the homotopy classification of graded categorical groups and their homomorphisms to develop a theory of extensions of categorical groups when a fixed group of operators is acting. For this kind of extensions we show a suitable Schreier’s theory and a precise theorem of classification, including obstruction theory, which generalizes both known results when the group of operators is trivial (categorical group extensions theory) or when the involved categorical groups are discrete (equivariant group extensions theory).

Idempotent‐separating extensions of regular semigroups

For a regular biordered set E, the notion of E‐diagram and the associated regular semigroup was introduced in our previous paper (1995). Given a regular biordered set E, an E‐diagram in a category C is a collection of objects, indexed by the elements of E and morphisms of C satisfying certain compatibility conditions. With such an E‐diagram A we associate a regular semigroup RegE(A) having E as its biordered set of idempotents. This regular semigroup is analogous to automorphism group of a group. This paper provides an application of RegE(A) to the idempotent‐separating extensions of regular semigroups. We introduced the concept of crossed pair and used it to describe all extensions of a regular semigroup S by a group E‐diagram A. In this paper, the necessary and sufficient condition for the existence of an extension of S by A is provided. Also we study cohomology and obstruction theories and find a relationship with extension theory for regular semigroups.

Pathogenesis of Chronic Pancreatitis: An Evidence-Based Review of Past Theories and Recent Developments

In the past several decades, four prominent theories of chronic pancreatitis pathogenesis have emerged: the toxic-metabolic theory, the oxidative stress hypothesis, the stone and duct obstruction theory, and the necrosis-fibrosis hypothesis. Although these traditional theories are formulated based on compelling scientific observations, substantial contradictory data also exist for each. Furthermore, the basic premises of some of these theories are directly contradictory. Because of the recent scientific progress in the underlying genetic, cellular, and molecular pathophysiology, there have been substantial advances in the understanding of chronic pancreatitis pathogenesis. This paper will provide an evidence-based review and critique of the traditional pathogenic theories, followed by a discussion of the new advances in pancreatic fibrogenesis. Moreover, we will discuss plausible pathogenic sequences applied to each of the known etiologies.

EULER OBSTRUCTION AND DEFECTS OF FUNCTIONS ON SINGULAR VARIETIES

Several authors have proved Lefschetz type formulas for the local Euler obstruction. In particular, a result of this type has been proved that turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of the paper is to determine what prevents the local Euler obstruction from satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or ‘defect’) of such functions. An interpretation of this defect is given in terms of vanishing cycles, which allows it to be calculated algebraically. When the function has an isolated singularity, the invariant can be defined geometrically, via obstruction theory. This invariant unifies the usual concepts of the Milnor number of a function and the local Euler obstruction of an analytic set.

D-branes on Calabi–Yau Manifolds and Superpotentials

We show how to compute terms in an expansion of the world-volume superpotential for fairly general D-branes on the quintic Calabi-Yau using linear sigma model techniques, and show in examples that this superpotential captures the geometry and obstruction theory of bundles and sheaves on this Calabi-Yau.

A lower bound to the action dimension of a group

The action dimension of a discrete group G, actdim(G), is defined to be the smallest integer m such that G admits a properly discontinuous action on a contractible m-manifold. If no such m exists, we define actdim(G) = infty. Bestvina, Kapovich, and Kleiner used Van Kampen's theory of embedding obstruction to provide a lower bound to the action dimension of a group. In this article, another lower bound to the action dimension of a group is obtained by extending their work, and the action dimensions of the fundamental groups of certain manifolds are found by computing this new lower bound.