The Moore space , where is an abelian group and , is a topological space which has non-trivial (reduced) homology group precisely only in dimension .
(This is somewhat dual to the notion of Eilenberg-MacLane space, which instead has nontrivial homotopy group in one single dimension.)
(the following is based on Hatcher)
Let be an abelian group. Take a presentation of i.e. a short exact sequence
where a free abelian group. Since , it is a free abelian group as well and we choose bases for and for . We then construct a CW complex
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 .
(Non-)Functoriality of the construction
The construction above is not functorial in because of the choice of bases (see more below). However, it does give a functor to the homotopy category .
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:
Carlsson provides counter examples for such “equivariant Moore spaces” for all non-cyclic groups.
There is thus no functor Ab Top that lifts since if there was such, it would induce, for any group a functor and in particular a positive answer to the Steenrod conjecture.
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 .
Co-Moore spaces are the Eckmann–Hilton duals of Eilenberg–Mac Lane spaces.
According to Baues, Moore spaces are -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.
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.
Gunnar Carlsson “A counterexample to a conjecture of Steenrod” Invent. Math. 64 (1981), no. 1, 171–174.