Shapes of compacta and ANR-systems

Mardešić, Sibe; Segal, Jack

doi:10.4064/fm-72-1-41-59

Cited by 119 publications

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“…Shape theory was later developed for more general spaces. Of special importance is the Mardesic-Segal approach [16] where shape theory is based on inverse systems of ANRs. In this approach, shapes are defined for arbitrary Hausdorff compacta.…”

Section: Introductionmentioning

confidence: 99%

An Intrinsic Description of Shape

Sanjurjo¹

1992

Transactions of the American Mathematical Society

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Abstract.We give in this paper a description of the shape category of compacta in terms of multivalued maps. We introduce the notion of a multi-net and prove that the shape category of compacta is isomorphic to the category HN whose objects are metric compacta and whose morphisms are homotopy classes of multi-nets. This description is intrinsic in the sense that it does not make use of external elements such as ANR-expansions or embeddings in appropriate AR-spaces. We present many applications of this new formulation of shape. IntroductionShape theory was introduced in 1968 by K. Borsuk [3] as a new classification of compact metric spaces which takes into account their global properties and neglects the local ones. Borsuk's main notion was that of a fundamental sequence which plays the same role as the notion of mapping in homotopy theory. Fundamental sequences are defined in terms of the neighborhoods of compacta embedded in a convenient AR-space. Shape theory was later developed for more general spaces. Of special importance is the Mardesic-Segal approach [16] where shape theory is based on inverse systems of ANRs. In this approach, shapes are defined for arbitrary Hausdorff compacta. Maps between such systems are defined as well as a notion of homotopy of such maps. This homotopy relation classifies maps between ANR-systems and these classes are called shape maps. In both cases the description of shape involves some external elements like ambient spaces where to embed the compacta or ANR-expansions.The aim of this paper is to present a new description of the shape category of compacta in which no use is made of such external elements. Our description is intrinsic in the sense that we only use maps (or sequences of maps) between compacta as opposed to the usual descriptions where maps are defined between some objects associated to the compacta. Our maps are multivalued. We introduce the notion of a multi-net between compacta which consists of a sequence of upper-semicontinuous multivalued maps progressively smaller, becoming more and more similar to a single-valued map. This notion is related to

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Section: Introductionmentioning

confidence: 99%

An Intrinsic Description of Shape

Sanjurjo¹

1992

Transactions of the American Mathematical Society

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“…This theory agrees with the singular one when the underlying space is an ANR (for information about them see [3,16]), in particular a differentiable manifold. Finally, the essentials of shape theory are contained in [5] or, for more exhaustive information, [11,18,19] and the books [17,20].…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

Shape properties of the boundary of attractors

Sánchez-Gabites¹

2007

Glas. Mat. Ser. III

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Dedicated to Sibe Mardešić in the occasion of his 80 th anniversary and in recognition of his guidance in the realm of geometric topology and shape theory Abstract. Let M be a locally compact metric space endowed with a continuous flow ϕ : M × R → M . Assume that K is a stable attractor for ϕ and P ⊆ A(K) is a compact positively invariant neighbourhood of K contained in its basin of attraction. Then it is known that the inclusion K → P is a shape equivalence and the question we address here is whether there exists some relation between the shapes of ∂K and ∂P . The general answer is negative, as shown by example, but under certain hypotheses on K the shape domination Sh(∂K) ≥ Sh(∂P ) or even the equality Sh(∂K) = Sh(∂P ) hold. However we also put under study interesting situations where those hypotheses are not satisfied, albeit other techniques such as Lefschetz's duality render results relevant to our question.

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“…These are a class of spaces more general than ANR's but less pathological than arbitrary compact metric spaces. If X is a movable metric compactum, then {h^(Xj),Pjt} satisfies M-L for any CNS X [36]. Thus X movable implies that hn(X) = h"(X).…”

Section: Proposition (22) Hn(point) -Ofor All Nmentioning

confidence: 99%

K-Theory and Steenrod Homology: Applications to the Brown-Douglas-Fillmore Theory of Operator Algebras

Kaminker¹,

Schochet²

1977

Transactions of the American Mathematical Society

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Abstract. The remarkable work of L. G. Brown, R. Douglas and P. Fillmore on operators with compact self-commutators once again ties together algebraic topology and operator theory. This paper gives a comprehensive treatment of certain aspects of that connection and some adjacent topics. In anticipation that both operator theorists and topologists may be interested in this work, additional background material is included to facilitate access. Introduction. L. G. Brown, R. Douglas and P. Fillmore [14], [15], [16](referred to as BDF) have forged a new link between operator theory and algebraic topology. Given a bounded operator £ G £ with £* £ -££* G %, the compact operators, they define an element [£] lying in an abelian group S\t(X), where X = a^rT) C C is the spectrum of the projection of £into the Calkin algebra 31 = £/9C. This class is zero if and only if £ is a compact perturbation of a normal operator. BDF then prove that &xl(X)=¿É°(C-X),

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Shapes of compacta and ANR-systems

Cited by 119 publications

References 0 publications

An Intrinsic Description of Shape

An Intrinsic Description of Shape

Shape properties of the boundary of attractors

K-Theory and Steenrod Homology: Applications to the Brown-Douglas-Fillmore Theory of Operator Algebras

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