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.

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 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).

See also

• Eilenberg–MacLane space, the homotopy analog.

References

• Hatcher, Allen. Algebraic topology, Cambridge University Press (2002), ISBN 0-521-79540-0. For further discussion of Moore spaces, see Chapter 2, Example 2.40. A free electronic version of this book is available on the author's homepage.