Cohomological dimension and approximate limits

Abstract. We shall prove a type of Mardešić factorization theorem for extension theory over an arbitrary stratum of CW-complexes in the class of arbitrary compact Hausdorff spaces. Our result provides that the space through which the factorization occurs will have the same strong countability property (e.g., strong countable dimension) as the original one had. Taking into consideration the class of compact Hausdorff spaces, this result extends all previous ones of its type. Our factorization theorem will simultaneously include factorization for weak infinite-dimensionality and for Property C, that is, for C-spaces.A corollary to our result will be that for any weight α and any finitely homotopy dominated CW-complex K, there exists a Hausdorff compactum X with weight wX ≤ α and which is universal for the property Xτ K and weight ≤ α. The condition Xτ K means that for every closed subset A of X and every map f : A → K, there exists a map F : X → K which is an extension of f . The universality means that for every compact Hausdorff space Y whose weight is ≤ α and for which Y τK is true, there is an embedding of Y into X.We shall show, on the other hand, that there exists a CW-complex S which is not finitely homotopy dominated but which has the property that for each weight α, there exists a Hausdorff compactum which is universal for the property Xτ S and weight ≤ α.

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Resolutions for metrizable compacta in extension theory

Rubin

Schapiro

2005

Trans. Amer. Math. Soc.

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Abstract. We prove a K-resolution theorem for simply connected CWcomplexes K in extension theory in the class of metrizable compacta X. This means that if K is a connected CW-complex, G is an abelian group, n ∈ N ≥2 , G = π n (K), π k (K) = 0 for 0 ≤ k < n, and extdim X ≤ K (in the sense of extension theory, that is, K is an absolute extensor for X), then there exists a metrizable compactum Z and a surjective map π : Z → X such that: This implies the G-resolution theorem for arbitrary abelian groupsIf in addition π n+1 (K) = 0, then (a) can be replaced by the stronger statement, (aa) π is K-acyclic.To say that a map π is K-acyclic means that for each x ∈ X, every map of the fiber π −1 (x) to K is nullhomotopic.

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Cohomological dimension and approximate limits

Cited by 5 publications

References 8 publications

Unbounded sets of maps and compactification in extension theory

Unbounded sets of maps and compactification in extension theory

The Mardesic factorization theorem for extension theory and c-separation

Resolutions for metrizable compacta in extension theory

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