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Acyclic space — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Acyclic_space
License (as archived): CC-BY-SA-4.0
Archived: 2026-11-07T12:14:37.424Z
SHA-256 of the archived text: 7521964fee9520476bee4518436d69f1554c3b87d158faa5e14f787ffce11933
Stable URL of this page: https://metraipolis.com/documento/dc_bbf4d8cfacdc

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Acyclic space — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Acyclic_space (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_bbf4d8cfacdc (Metraipolis archive dc_bbf4d8cfacdc, sha256 7521964fee9520476bee4518436d69f1554c3b87d158faa5e14f787ffce11933)

BibTeX

@misc{dcbbf4d8cfacdc,
  title = {Acyclic space — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Acyclic_space},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_bbf4d8cfacdc; sha256 7521964fee9520476bee4518436d69f1554c3b87d158faa5e14f787ffce11933},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Acyclic_space. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In mathematics, an acyclic space is a nonempty topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of X are isomorphic to the corresponding homology groups of a point. In other words, using the idea of reduced homology, H ~ i ( X ) = 0 , ∀ i ≥ − 1. {\displaystyle {\tilde {H}}_{i}(X)=0,\quad \forall i\geq -1.} It is common to consider such a space as a nonempty space without "holes"; for example, a circle or a sphere is not acyclic but a disc or a ball is acyclic. This condition however is weaker than asking that every closed loop in the space would bound a disc in the space, all we ask is that any closed loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface." The condition of acyclicity on a space X implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of X to the circle or to the higher spheres is null-homotopic. If a space X is contractible, then it is also acyclic, by the homotopy invariance of homology. The converse is not true, in general. Nevertheless, if X is an acyclic CW complex, and if the fundamental group of X is trivial, then X is a contractible space, as follows from the Whitehead theorem and the Hurewicz theorem.

Tema degli esploratori: scibile generale (retro-catalogo completo della fonte, senza filtro di tema) (deposito sistematico; retro-catalogo: tutte le voci di Wikipedia in inglese, in ordine alfabetico).
Fonte: Wikipedia in inglese, voce «Acyclic space» — https://en.wikipedia.org/wiki/Acyclic_space
Wikidata Q4677988 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q4677988
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Acyclic_space?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1249237472, consultata il 2026-11-07T12:09:36.067Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: cc4486483ef6186391b5bd7e327b19120a0bde5b1db1b4c91b26cf2e7dcd0d13