Homotopy theory | Homology theory | Algebraic topology

Acyclic space

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. (Wikipedia).

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Related pages

Topological space | Aspherical space | Group extension | Hurewicz theorem | Homotopy group | Binary icosahedral group | 3-manifold | Topology | CW complex | Plus construction | Classifying space | Superperfect group | Reduced homology | Homology (mathematics) | Mathematics | Perfect group | Fundamental group | Homology sphere | Manifold | Whitehead theorem | Commutator subgroup | Simplicial complex | Contractible space