Acyclic space
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In mathematics, an acyclic space is a topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that the 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,
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.
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[edit] Examples
Acyclic spaces occur in topology, where they can be used to construct other, more interesting topological spaces.
For example, if one removes a single point from a manifold M which is a homology sphere, one gets such a space. The homotopy groups of an acyclic space X do not vanish in general, because the fundamental group need not be trivial. For example, the punctured Poincaré sphere is an acyclic, 3-dimensional manifold which is not contractible.
This gives a repertoire of examples, since the first homology group is the abelianisation of the fundamental group. With every perfect group G one can associate a (canonical, terminal) acyclic space, whose fundamental group is a central extension of the given group G.
The homotopy groups of these associated acyclic spaces are closely related to Quillen's plus construction on the classifying space BG.
[edit] Acyclic groups
An acyclic group is a group G whose classifying space BG is acyclic. Every acyclic group is a perfect group (in fact, a superperfect group), but neither implication can be reversed.
[edit] References
- Emmanuel Dror, "Acyclic spaces", Topology 11 (1972), 339–348. MR0315713
- Emmanuel Dror, "Homology spheres", Israel Journal of Mathematics 15 (1973), 115–129. MR0328926
- A. Jon Berrick and Jonathan A. Hillman, "Perfect and acyclic subgroups of finitely presentable groups", Journal of the London Mathematical Society (2) 68 (2003), no. 3, 683–698. MR2009444
