Moore space (algebraic topology)

In algebraic topology, a branch of mathematics, Moore space is the name given to a particular type of topological space that is the homology analogue of the Eilenberg–Maclane spaces of homotopy theory, in the sense that it has only one nonzero homology (rather than homotopy) group.

Formal definition
Given an abelian group G and an integer n ≥ 1, let X be a CW complex such that


 * $$H_n(X) \cong G$$

and


 * $$\tilde{H}_i(X) \cong 0$$

for i ≠ n, where $$H_n(X)$$ denotes the n-th singular homology group of X and $$\tilde{H}_i(X)$$ is the i-th reduced homology group. Then X is said to be a Moore space. Some authors also require that X be simply-connected if n>1.

Examples

 * $$S^n$$ is a Moore space of $$\mathbb{Z}$$ for $$n\geq 1$$.
 * $$\mathbb{RP}^2$$ is a Moore space of $$\mathbb{Z}/2\mathbb{Z}$$ for $$n=1$$.