The S_n-equivalence of compacta

Abstract

Abstract. By reducing the Mardešić S-equivalence to a finite case, i.e., to each n ∈ {0} ∪ N separately, we have derived the notions of Snequivalence and S n+1 -domination of compacta. The Sn-equivalence for all n coincides with the S-equivalence. Further, the S n+1 -equivalence implies S n+1 -domination, and the S n+1 -domination implies Sn-equivalence. The S 0 -equivalence is a trivial equivalence relation, i.e., all non empty compacta are mutually S 0 -equivalent. It is proved that the S 1 -equivalence is strictly finer than the S 0 -equivalence, and that the S 2 -equivalence is strictly finer than the S 1 -equivalence. Thus, the S-equivalence is strictly finer than the S 1 -equivalence. Further, the S 1 -equivalence classifies compacta which are homotopy (shape) equivalent to ANR's up to the homotopy (shape) types. The S 2 -equivalence class of an FANR coincides with its S-equivalence class as well as with its shape type class. Finally, it is conjectured that, for every n, there exists n > n such that the S n -equivalence is strictly finer than the Sn-equivalence.

The induced homology and homotopy functors on the coarse shape category

“…This means that (g, h λ ) is a morphism of inv-C. Thus, it represents a morphisms h : Z → N of pro-C, and because of (14), holds (12), which completes the proof.…”

“…In the last few years several articles have been published in which are introduced some new classifications of metric compacta (see [11,12]) and even classifications of all topological spaces (see [5,13]) which are coarser than the shape type classification. These classifications are playing an important role because they provide better information about spaces having different shape types than the standard shape classification.…”

The coarse shape

“…So N. Uglešić [20] studied the Borsuk's quasi-equivalence and quasi-affinity and introduced some new ones, Mardešić and Uglešić [16] described the S * -equivalence (a uniformization of the S-equivalence) in a category framework, Uglešić and B.Červar [21,22] derived the S n -equivalences, n ∈ N, from the S-equivalence and constructed a categorical subshape spectrum for compacta, while A. Kadlof, N. Koceić Bilan and Uglešić [8] proved (solved the problem stated in [3]) that the Borsuk quasi-equivalence is not transitive.…”

Metrization of pro-morphism sets

“…By [23], Example 2. n i ) be any representatives of f n and g n respectively, n ∈ N. Notice that every homotopy commutative diagram relating X to Y and vice versa must be (strictly) commutative, and that all the mappings f n j and g n i must be surjective. Then, a straightforward analysis (compare the proof following [23], Example 2.9) shows that, for every n ∈ N, the inequality f n (1) ≥ 2n + 1 must be satisfied. Consequently, there is no unique index function for any sequence ((f n , f n j )) representing (f n ).…”

“…Especially, if Y ∈ ObD, then every shape morphism φ : X → Y , i.e., every f : X → Y of tow-D, admits a unique representative f : X → Y of C. The most interesting example is C = HcM (the homotopy category of metrizable compacta) and D = HcAN R (the homotopy category of compact ANR's) or D = HcP ol (the homotopy category of compact polyhedra). We hereby also want to involve in our considerations the S-equivalence ( [11,12]) and S * -equivalence ( [14]) (as well as the S n -and S + n -equivalence of [23] and [5]). These equivalences and corresponding dominations are well defined in every category tow-A.…”