# Moore space (algebraic topology)

See also Moore space for other meanings in mathematics.

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.

## 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 in, where Hn(X) denotes the n-th singular homology group of X and $\tilde{H}_i(X)$ is the ith reduced homology group. Then X is said to be a Moore space.

## Examples

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