TY - JOUR

T1 - Cohomological dimension and acyclic resolutions

AU - Koyama, Akira

AU - Yokoi, Katsuya

N1 - Funding Information:
*Current address: Department of Mathematics and Computer Science, Interdisciplinary Faculty of Science and Engineering, Shimane University, Matsue, 690-8504, Japan. E-mail addresses: koyama@cc.osaka-kyoiku.ac.jp (A. Koyama), yokoi@math.shimane-u.ac.jp (K. Yokoi). 1The second author was partially supported by the University of Tsukuba Research Projects and the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists (No. 11740035, 1999–2000).

PY - 2002/5/15

Y1 - 2002/5/15

N2 - Let G be an Abelian group admitting a homomorphism α: ℤ→G such that the induced homomorphisms α⊗id: ℤ⊗G→G⊗G and α*: Hom(G,G)→Hom(ℤ,G) are isomorphisms. We show that for every simplicial complex L there exists an Edwards-Walsh resolution ω: EWG(L,n)→ L . As applications of it we give several resolution theorems. In particular, we have Theorem. Let G be an arbitrary Abelian group. For every compactum X with c-dimGX≤n there exists a G-acyclic map f: Z→X from a compactum Z with dimZ≤n+2 and c-dimGZ≤n+1. Our methods determine other results as well. If the group G is cyclic, then one can obtain Z with dimZ≤n. In certain other cases, depending on G, we may resolve X in such a manner that dimZ≤n+1 and c-dimGZ≤n.

AB - Let G be an Abelian group admitting a homomorphism α: ℤ→G such that the induced homomorphisms α⊗id: ℤ⊗G→G⊗G and α*: Hom(G,G)→Hom(ℤ,G) are isomorphisms. We show that for every simplicial complex L there exists an Edwards-Walsh resolution ω: EWG(L,n)→ L . As applications of it we give several resolution theorems. In particular, we have Theorem. Let G be an arbitrary Abelian group. For every compactum X with c-dimGX≤n there exists a G-acyclic map f: Z→X from a compactum Z with dimZ≤n+2 and c-dimGZ≤n+1. Our methods determine other results as well. If the group G is cyclic, then one can obtain Z with dimZ≤n. In certain other cases, depending on G, we may resolve X in such a manner that dimZ≤n+1 and c-dimGZ≤n.

KW - Acyclic resolution

KW - Cohomological dimension

KW - Edwards-Walsh resolution

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U2 - 10.1016/S0166-8641(01)00015-3

DO - 10.1016/S0166-8641(01)00015-3

M3 - Article

AN - SCOPUS:0038012812

SN - 0166-8641

VL - 120

SP - 175

EP - 204

JO - Topology and its Applications

JF - Topology and its Applications

IS - 1-2

ER -