Moore space (algebraic topology)

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

${\displaystyle H_{n}(X)\cong G}$

and

${\displaystyle {\tilde {H}}_{i}(X)\cong 0}$

for i ≠ n, where ${\displaystyle H_{n}(X)}$ denotes the n-th singular homology group of X and ${\displaystyle {\tilde {H}}_{i}(X)}$ 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.^{[citation needed]}

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

• Eilenberg–MacLane space, the homotopy analog.
• Homology sphere

• 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.