(the following is based on Hatcher)
0\rightarrow H \rightarrow F\rightarrow G\rightarrow 0
by taking the -th skeleton to be and for each we attach an -cell as follows:
Write and let be if and otherwise. Define an attaching map by contracting spheres in thus defining a map and then map each to by a degree .
The resulting CW-complex can be seen to have the desired properties via cellular homology.
The homotopy type of is determined by specifying and .
The functoriality problem of the construction above cannot be corrected. That is, there is no functor that lifts . This can be seen as a corollary of a counterexample of Carlsson which gives a negative answer to a conjecture of Steenrod:
Given a group a -module and a natural number , there is a -space which has only one non-zero reduced homology G-module in dimension that satisfy as -modules.
Carlsson provides counter examples for such “equivariant Moore spaces” for all non-cyclic groups.
Moreover, there can also not be an (∞,1)-functor that lifts since this will similarly yield an -functor where is the (∞,1)-category of ∞-actions of on spaces. Since there is a “rigidification” functor this would yield an (ordinary) functor which does not exist by our previous observation.
There is also a cohomology analogue known as a co-Moore space or a Peterson space, but this is not defined for all abelian . Spheres are both Moore and co-Moore spaces for .
According to Baues, Moore spaces are -duals to Eilenberg–Mac Lane spaces. This leads to an extensive duality for connected CW complexes.
Hans J. Baues, Homotopy types, in Handbook of Algebraic Topology, (edited by I.M. James), North Holland, 1995.
Gunnar Carlsson “A counterexample to a conjecture of Steenrod” Invent. Math. 64 (1981), no. 1, 171–174.