For an abelian group and an integer , a Moore space is a CW complex whose reduced homology is in degree and zero in every other degree.
Attaching a two-cell to by a map of degree gives the Moore space . Its positive-dimensional cellular chain complex isso and all its other reduced integral homology groups vanish.
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In algebraic topology, a **Moore space** refers to a particular type of topological space that arises in the study of homotopy theory and is used in the construction of certain types of homotopy groups and CW complexes. A Moore space is defined as a connected space \( M(X, n) \) that has the following properties: 1. **Construction**: The space is constructed from a space \( X \) and a positive integer \( n \).