The Complement Theorem in n-Shape Theory for Compact Pairs

Yūji Akaike and Katsuro Sakai, Tsukuba, Japan

AMS Subject classification: 54C56, 57N25, 55P55

Keywords: The complement theorem, n-shape of compact pairs, μ^{n+1}-manifold pair, Z-pair, the Z-pair approximation theorem, the Z-pair unknotting theorem, the relative n-homotopy extension theorem

Abstract: Let μ^{n+1} be the (n + 1)-dimensional universal Menger compactum and μ_{0}^{n+1} a Z-set in μ^{n+1} which is homeomorphic (≈) to μ^{n+1}. In this paper, we generalize Chigogidze’s Complement Theorem to compact pairs as follows: for any two Z-pairs (X, X_{0}) and (Y, Y_{0}) in (μ^{n+1}, μ_{0}^{n+1}), n−Sh(X, X_{0}) = n−Sh(Y, Y_{0}) if and only if (μ^{n+1} \ X, μ_{0}^{n+1} \ X_{0}) ≈ (μ^{n+1} \ Y, μ_{0}^{n+1} \ Y_{0}). To prove this, we generalize two theorems on μ^{n+1}-manifolds (the Z-set approximation theorem and the Z-set unknotting theorem) to μ^{n+1}-manifold pairs.