The theory ofn-shapes

Menger (Nöbeling) Manifolds versus Hilbert Cube (Space) Manifolds - A Categorical Comparison

“…First of all we note that the (n + 1)-dimensional universal Menger compactum µ n+1 also admits [9] a Z-skeletoid Σ n+1 such that its complement ν n+1 = µ n+1 − Σ n+1 is homeomorphic [10], [7,Theorem 5.5.5] to the (n + 1)-dimensional universal Nöbeling space N 2n+3 n+1 . Now we proceed as above.…”

“…Two maps f, g : X → Y are said to be n-homotopic n ≥ 0 (notation: f n ≃ g) if for any map h : Z → X, defined on an at most n-dimensional space Z, the compositions f • h and g • h are homotopic in the usual sense. The reader can find basic properties of n-homotopic maps in [14], [2], [7], [9].…”

Locally n-Connected Compacta and UVn -Maps

“…So, according to this corollary, X is the limit space of a σ-complete inverse system consisting of metric cell-like compacta. In case X is an UV n -compactum, it has an n-shape of a point (this notion was introduced by Chigogidze in [4]), and the above arguments apply.…”

On stable cohomotopy groups of compact spaces II

“…Let Sh S ∞ n,m = Sh (T ,P) denote the abstract shape category for the pair (T , P) = (S ∞ n+1 (Comp), S ∞ n+1 (CW m )), where n m. The case m = −1 and the related category Sh S ∞ n,−1 is complementary to the n-shape theory ( [1,6] or [5]). However, it seems that the case m = n is crucial to applications and we are planing to concentrate our attention on this case and on the stable ∞ n -shape category…”