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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,

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rdfs:label	Acyclic space (en)
rdfs:comment	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, (en)
dcterms:subject	Homology theory Algebraic topology Homotopy theory
Wikipage page ID	10355069 (xsd:integer)
Wikipage revision ID	1078346090 (xsd:integer)
Link from a Wikipage to another Wikipage	Whitehead theorem Mathematics Classifying space Contractible space Commutator subgroup Fundamental group Perfect group Superperfect group Homology theory Topology 3-manifold Algebraic topology Daniel Quillen Israel Journal of Mathematics Simplicial complex Homotopy theory Group homology Hurewicz theorem Binary icosahedral group Homology (mathematics) Homology sphere Homotopy group Topology (journal) Reduced homology Aspherical space Manifold CW complex Group extension Journal of the London Mathematical Society Plus construction Topological space Poincaré homology sphere
sameAs	Acyclic space Acyclic space Acyclic space
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Citation dbt:Technical
id	p/a110270 (en)
title	Acyclic groups (en)
has abstract	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, 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 discor 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. (en)
prov:wasDerivedFrom	wikipedia-en:Acyclic_space?oldid=1078346090&ns=0
page length (characters) of wiki page	4137 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Acyclic_space
is Link from a Wikipage to another Wikipage of	Volodin space Q-construction Superperfect group Acyclic group Aspherical space Acyclic spaces
is Wikipage redirect of	Acyclic group Acyclic spaces
is foaf:primaryTopic of	wikipedia-en:Acyclic_space

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