# Cohomological dimension and acyclic resolutions

Akira Koyama, Katsuya Yokoi

8 引用 (Scopus)

### 抄録

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.

元の言語 English 175-204 30 Topology and its Applications 120 1-2 https://doi.org/10.1016/S0166-8641(01)00015-3 Published - 2002 5 15 Yes

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Compactum
Cohomological Dimension
Abelian group
Simplicial Complex
Homomorphisms
Theorem
Homomorphism
Resolve
Isomorphism
Arbitrary

### ASJC Scopus subject areas

• Geometry and Topology

### これを引用

Cohomological dimension and acyclic resolutions. / Koyama, Akira; Yokoi, Katsuya.

：: Topology and its Applications, 巻 120, 番号 1-2, 15.05.2002, p. 175-204.

Koyama, Akira ; Yokoi, Katsuya. / Cohomological dimension and acyclic resolutions. ：: Topology and its Applications. 2002 ; 巻 120, 番号 1-2. pp. 175-204.
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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.

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