Isotopy invariants of topological spaces

1960 ◽  
Vol 255 (1282) ◽  
pp. 331-366 ◽  
Keyword(s):  
Homotopy Type ◽  
Algebraic Topology ◽  
Complete System ◽  
Topological Spaces ◽  
Recent Developments ◽  
The Difference ◽  
Linear Graphs ◽  
Homotopy Invariants ◽  
General Method

A few decades ago, topologists had already emphasized the difference between homotopy and isotopy. However, recent developments in algebraic topology are almost exclusively on the side of homotopy. Since a complete system of homotopy invariants has been obtained by Postnikov, it seems that hereafter we should pay more attention to isotopy invariants and new efforts should be made to attack the classical problems. The purpose of this paper is to introduce and study new algebraic isotopy invariants of spaces. A general method of constructing these invariants is given by means of a class of functors called isotopy functors. Special isotopy functors are constructed in this paper, namely, the m th residual functor R m and the m th. enveloping functor E m . Applications of these isotopy invariants to linear graphs are given in the last two sections. It turns out that these invariants can distinguish various spaces belonging to the same homotopy type.

Suspension of the Lusternik-Schnirelmann Category

1970 ◽  
Vol 22 (6) ◽  
pp. 1129-1132
Author(s):  
William J. Gilbert
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Category Structure ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Cw Complexes ◽  
Lusternik Schnirelmann ◽  
Image Position ◽  
Simply Connected ◽  
Homotopy Invariants

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],We prove thatandIt is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

scholarly journals Russia’s Foreign Policy Evolution in the New Balkan Landscape

2020 ◽  
Vol 56 (3-4) ◽  
pp. 179-199
Author(s):  
Ekaterina Entina ◽  
Alexander Pivovarenko
Keyword(s):  
Foreign Policy ◽  
Soviet Union ◽  
Regional Security ◽  
Southeast Europe ◽  
The Balkans ◽  
Current Trends ◽  
The Soviet Union ◽  
Recent Developments ◽  
The Difference

The article reflects on the issue of the foreign policy strategy of modern Russia in the Balkans region. One of the most significant aspects of this problem is the difference in views between Russia and the West. Authors show how different interpretations of the events in former Yugoslavia in the 1990s and the beginning of the 2000s predetermined the sense of mutual suspicion and mistrust which spread to other regions such as the post-Soviet space. Exploring differences between the Russian and the Western (Euro-Atlantic) views on the current matters, authors draw attention to fundamental differences in terminology: while the Western narrative promotes more narrow geographical and political definitions (such as the Western Balkan Six), traditional Russian experts are more inclined to wider or integral definitions such as “the Balkans” and “Central and Southeast Europe”. Meanwhile none of these terms are applicable for analysis of the current trends such as the growing transit role of the Balkans region and its embedding in the European regional security architecture. Therefore, a new definition is needed to overcome the differences in vision and better understand significant recent developments in the region. Conceptualizing major foreign policy events in Central and Southeast Europe during the last three decades (the 1990s, 2000s and 2010s), authors demonstrate the significance of differences in tools and methods between the Soviet Union and the modern Russia. Permanent need for adaptation to changing political and security context led to inconsistence in Russian Balkan policy in the 1990s. Nevertheless, Russia was able to preserve an integral vision of the region and even to elaborate new transregional constructive projects, which in right political circumstances may promote stability and become beneficial for both Russia and the Euro-Atlantic community.

scholarly journals Criteria for components of a function space to be homotopy equivalent

2008 ◽  
Vol 145 (1) ◽  
pp. 95-106 ◽  
Author(s):  
GREGORY LUPTON ◽  
SAMUEL BRUCE SMITH
Keyword(s):  
Function Space ◽  
Homotopy Type ◽  
Homotopy Equivalent ◽  
General Method

AbstractWe give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

Homotopy invariants and continuous mappings

1950 ◽  
Vol 202 (1069) ◽  
pp. 253-263 ◽  
Keyword(s):  
Homotopy Type ◽  
Betti Numbers ◽  
Continuous Mappings ◽  
Numerical Invariants ◽  
Homotopy Invariants

This paper contributes new numerical invariants to the topology of a certain class of polyhedra. These invariants, together with the Betti numbers and coefficients of torsion, characterize the homotopy type of one of these polyhedra. They are also applied to the classification of continuous mappings of an ( n + 2)-dimensional polyhedron into an ( n + 1)-sphere ( n > 2).

CROSSED PRODUCT C*-ALGEBRAS AND ALGEBRAIC TOPOLOGY

1996 ◽  
Vol 08 (04) ◽  
pp. 623-637
Author(s):  
JUDITH A. PACKER
Keyword(s):  
Algebraic Topology ◽  
Equivariant Cohomology ◽  
Automorphism Groups ◽  
Crossed Product ◽  
Fiber Bundles ◽  
Topological Invariants ◽  
C Algebras ◽  
Recent Developments ◽  
Continuous Trace ◽  
Cech Cohomology

We discuss some recent developments that illustrate the interplay between the theory of crossed products of continuous trace C*-algebras and algebraic topology, summarizing results relating topological invariants coming from the theory of fiber bundles to continuous trace C*-algebras and their automorphism groups and the structure of the associated crossed product C*-algebras. This survey article starts from the classical theory of Dixmier, Douady, and Fell, and discusses the more recent work of Echterhoff, Phillips, Raeburn, Rosenberg, and Williams, among others. The topological invariants involved are Čech cohomology, the cohomology of locally compact groups with Borel cochains of C. Moore, and the recently introduced equivariant cohomology theory of Crocker, Kumjian, Raeburn and Williams.

Comparison of Regional Lung Deformation Between Dynamic and Static CT Imagery Using Inverse Consistent Registration

2009 ◽  
Author(s):  
Ryan Amelon ◽  
Kai Ding ◽  
Kunlin Cao ◽  
Gary E. Christensen ◽  
Joseph M. Reinhardt ◽  
...  
Keyword(s):  
Image Data ◽  
Lung Mechanics ◽  
Ct Scans ◽  
Positive Pressure ◽  
Displacement Fields ◽  
Regional Patterns ◽  
Recent Developments ◽  
The Difference ◽  
Key Aspects ◽  
Lung Deformation

The mechanics of lung deformation is traditionally assessed at a whole-lung or lobar level. We submit that key aspects of lung mechanics maybe better understood by studying regional patterns of lung deformation by leveraging recent developments in tomographic imaging and image processing techniques. Our group has developed an inverse consistent registration technique for estimating local displacement distributions from paired lung CT volumes [1,2]. This facilitates the estimation of strain distributions and consequently, the regional patterns in volume change and its preferential directionalities (anisotropy in deformation). In this study, we use this novel method to compare regional deformation in the lungs between static and dynamic inflations in an adult sheep. Much of our research has focused on registration of static lung images at different positive end-expiratory pressures (PEEP). More recently, respiratory-gated CT scans of supine, positive-pressure inflated sheep lungs have been gathered in order to compare the displacement fields of a dynamically inflating lung to the static lung scans. The theory is that scanning a dynamically inflating lung will more accurately reflect natural deformation during breathing by realizing time-dependent mechanical properties (viscoelasticity). The downside to human dynamic lung imaging is the increased radiation dose necessary to acquire the image data across the respiratory cycle, though low-dose CT scans are an option [3]. This experiment observed the difference in strain distribution between dynamically inflated lungs versus static apneic lungs using the inverse consistent image registration developed in our lab.

Cost–benefit and water resources policy: a survey

Water Policy ◽  
2011 ◽  
Vol 14 (2) ◽  
pp. 250-280 ◽  
Author(s):  
Frank A. Ward
Keyword(s):  
Water Policy ◽  
Cost Benefit Analysis ◽  
Cost Benefit ◽  
Benefit Analysis ◽  
Trade Offs ◽  
Wide Range ◽  
Recent Developments ◽  
The Difference ◽  
Policy Cost ◽  
The Right

This paper reviews recent developments in cost–benefit analysis for water policy researchers who wish to understand the applications of economic principles to inform emerging water policy debates. The cost–benefit framework can provide a comparison of total economic gains and losses resulting from a proposed water policy. Cost–benefit analysis can provide decision-makers with a comparison of the impacts of two or more water policy options using methods that are grounded in time-tested economic principles. Economic efficiency, measured as the difference between added benefits and added costs, can inform water managers and the public of the economic impacts of water programs to address peace, development, health, the environment, climate and poverty. Faced by limited resources, cost–benefit analysis can inform policy choices by summarizing trade-offs involved in designing, applying, or reviewing a wide range of water programs. The data required to conduct a cost–benefit analysis are often poor but the steps needed to carry out that analysis require posing the right questions.

Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem

1993 ◽  
Vol 114 (1) ◽  
pp. 163-189 ◽  
Author(s):  
M. Bullejos ◽  
P. Carrasco ◽  
A. M. Cegarra
Keyword(s):  
Abelian Group ◽  
Homotopy Type ◽  
Abelian Groups ◽  
Group Structure ◽  
Cohomology Theory ◽  
Classification Theorem ◽  
Topological Spaces ◽  
Simplicial Sets

AbstractIn this paper we use Takeuchy–Ulbrich's cohomology of complexes of categories with abelian group structure to introduce a cohomology theory for simplicial sets, or topological spaces, with coefficients in symmetric cat-groups . This cohomology is the usual one when abelian groups are taken as coefficients, and the main topological significance of this cohomology is the fact that it is equivalent to the reduced cohomology theory defined by a Ω-spectrum, {}, canonically associated to . We use the spaces to prove that symmetric cat-groups model all homotopy type of spaces X with Πi(X) = 0 for all i ╪ n, n + 1 and n ≥ 3, and then we extend Eilenberg–MacLane's classification theorem to those spaces: .

scholarly journals Topological pattern recognition for point cloud data

Acta Numerica ◽  
2014 ◽  
Vol 23 ◽  
pp. 289-368 ◽  
Author(s):  
Gunnar Carlsson
Keyword(s):  
Pattern Recognition ◽  
Algebraic Topology ◽  
Point Cloud ◽  
Metric Spaces ◽  
Persistent Homology ◽  
Topological Spaces ◽  
Data Sets ◽  
Point Cloud Data ◽  
Cloud Data ◽  
Finite Metric Spaces

In this paper we discuss the adaptation of the methods of homology from algebraic topology to the problem of pattern recognition in point cloud data sets. The method is referred to aspersistent homology, and has numerous applications to scientific problems. We discuss the definition and computation of homology in the standard setting of simplicial complexes and topological spaces, then show how one can obtain useful signatures, called barcodes, from finite metric spaces, thought of as sampled from a continuous object. We present several different cases where persistent homology is used, to illustrate the different ways in which the method can be applied.

RACK SPACES AND LOOP SPACES

1999 ◽  
Vol 08 (01) ◽  
pp. 99-114 ◽  
Author(s):  
Bert Wiest
Keyword(s):  
Homotopy Type ◽  
Loop Space ◽  
Topological Spaces ◽  
Loop Spaces

We prove that the rack and quandle spaces of links in 3-manifolds, considered only as topological spaces (disregarding their cubical structure), are closely related to certain subspaces of the loop spaces on the 3-manifold, which we call the vertical and the straight loop space of the link. Using these models we prove that the homotopy type of the non-augmented rack and quandle spaces of a link L in a 3-manifold M depends essentially only on the homotopy type of the pair (M,M -L).

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