Moore space (algebraic topology)

For other uses, see Moore space (disambiguation).

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).

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.

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Moore space (algebraic topology)

Formal definition

Examples

See also

References