### Abstract

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: EW_{G}(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-dim_{G}X≤n there exists a G-acyclic map f: Z→X from a compactum Z with dimZ≤n+2 and c-dim_{G}Z≤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-dim_{G}Z≤n.

Original language | English |
---|---|

Pages (from-to) | 175-204 |

Number of pages | 30 |

Journal | Topology and its Applications |

Volume | 120 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2002 May 15 |

Externally published | Yes |

### Fingerprint

### Keywords

- Acyclic resolution
- Cohomological dimension
- Edwards-Walsh resolution

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*120*(1-2), 175-204. https://doi.org/10.1016/S0166-8641(01)00015-3

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

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 120, no. 1-2, pp. 175-204. https://doi.org/10.1016/S0166-8641(01)00015-3

}

TY - JOUR

T1 - Cohomological dimension and acyclic resolutions

AU - Koyama, Akira

AU - Yokoi, Katsuya

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

UR - http://www.scopus.com/inward/record.url?scp=0038012812&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038012812&partnerID=8YFLogxK

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

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

M3 - Article

AN - SCOPUS:0038012812

VL - 120

SP - 175

EP - 204

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 1-2

ER -