nLab
Moore space

The Moore space $M(G,n)$, where $G$ is an abelian group and $n\ge 1$, is a space which has non-trivial (reduced) homology group $G$ only in dimension $n$. There is also a cohomology analogue known as a co-Moore space, but this is not defined for all abelian $G$. Spheres are both Moore and co-Moore spaces for $G=\mathbb{Z}$.

Co-Moore spaces are the Eckmann–Hilton duals of Eilenberg–Mac Lane spaces.

According to Baues, Moore spaces are $H\pi$-duals to Eilenberg–Mac Lane spaces. This leads to an extensive duality for connected CW complexes.

Just as there is a Postnikov decomposition of a space as a tower of fibrations, so there is a Moore decomposition? as a tower of cofibrations.

References

• Example 2.40 of Hatcher Algebraic Topology.
• Marek Golasinski? and Gonçalves, On Co-Moore Spaces
• Hans J. Baues, Homotopy types, in Handbook of Algebraic Topology, (edited by I.M. James), North Holland, 1995.