Moore space (algebraic topology)

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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[edit]

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[edit]

  • 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[edit]