2011/01/13 by Leonard R. Rubin, Rubin, Leonard R., Vera Tonić +1
Mathematics · #54F45 #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Primary: 55M10 #Secondary: 55P20 #math.AT #math.GN #math.GT #msc:54F45 #msc:55M10 #msc:55P20
paper · pdf · doi:10.48550/arxiv.1101.2480
18 pages, title change in version 3, old title: "Z/p-acyclic resolutions in the strongly countable Z/p-dimensional case"
arxiv created 2013/01/28 · arxiv updated 2013/01/29
We prove the following Theorem: Let X be a nonempty compact metrizable space, let l1 ≤ l2 ≤... be a sequence of natural numbers, and let X1 ⊂ X2 ⊂... be a sequence of nonempty closed subspaces of X such that for each k in N, dimZ/p Xk ≤ lk < ∞. Then there exists a compact metrizable space Z, having closed subspaces Z1 ⊂ Z2 ⊂..., and a surjective cell-like map π: Z → X, such that for each k in N, (a) dim Zk ≤ lk, (b) π(Zk) = Xk, and (c) π| Zk: Zk → Xk is a Z/p-acyclic map. Moreover, there is a sequence A1 ⊂ A2 ⊂... of closed subspaces of Z, such that for each k, dim Ak ≤ lk, π|Ak: Ak→ X is surjective, and for k in N, Zk⊂ Ak and π|Ak: Ak→ X is a UVlk-1-map. It is not required that X be the union of all Xk, nor that Z be the union of all Zk. This result generalizes the Z/p-resolution theorem of A. Dranishnikov, and runs parallel to a similar theorem of S. Ageev, R. Jiménez, and L. Rubin, who studied the situation where the group was Z.