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

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

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]}

Examples

• ${\displaystyle S^{n}$ 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 ${\displaystyle n=1}$.

See also

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

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.