Strong Shape and Homology

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.

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

Shape properties of the boundary of attractors

Sánchez-Gabites¹

2007

“…The differential operators in these complexes are defined in the same way as above and they are continuous in the corresponding topologies. If for any topological space X we consider an AN R-resolution in the sense of Mardešić ([12]), then the first construction defines the strong homologȳ H n (X; K) (for the theory of strong homology see [12]) andČech cohomology H n (X; K) of X with coefficients in a field K; if one considers the direct system of all compact Hausdorff subsets in X, the second construction presents classical homology with compact supports H c n (X; K) and the strong cohomologyH n (X; K) of X (for the theory of strong cohomology see [9]) with coefficients in a field K. Thus all questions which arose concern the topological vector space structures of classical (co)homologies of topological spaces with coefficients in the field K (R or C).…”

Section: P 158])mentioning

confidence: 99%

On hereditary reflexivity of topological vector spaces. The de Rham cochain and chain spaces

Lisica¹

2008

Abstract. For proving reflexivity of the spaces of de Rham cohomology and homology of C ∞ -manifolds the author considers the notion of hereditary reflexivity as well as the notion of dual hereditary reflexivity of locally convex topological vector spaces which is interesting in itself. Complete barrelled nuclear spaces with complete nuclear duals turn out to be hereditarily reflexive. The Pontryagin duality in locally convex topological spaces is also considered.

“…A direct system X : I → C gives an inverse system X : I op → C, and vice-versa. We refer to [8,7,5], for constructions and terminology concerning inverse and direct systems. Let Σ ⊂ Mor(C).…”

Section: Orthogonality and Saturationmentioning

confidence: 99%

Orthogonality, saturation and shape

Stramaccia

2007

Abstract. The class of shape equivalences for a pair (C, K) of categories is the orthogonal of K, that is Σ = K ⊥ . Then Σ is internally saturated (Σ = Σ ⊤⊥ ). On the other hand, every internally saturated class of morphisms Σ ⊂ Mor(C), is the class of shape equivalences for some pair (C, K). Moreover, every class of shape equivalences Σ enjoys a calculus of left fractions and such a fact allows one to use techniques from categories of fractions to obtain conditions for Σ ⊤ to be reflective or proreflective in C.