nLab Moore space

Idea

The Moore space $M(G, n)$, where $G$ is an abelian group and $n \geq 1$, is a topological space which has non-trivial (reduced) homology group $G$ precisely only in dimension $n$ and is simply connected if $n \geq 2$.

(This is somewhat dual to the notion of Eilenberg-MacLane space, which instead has nontrivial homotopy group in one single dimension.)

Definition

(the following is based on Hatcher)

Let $G$ be an abelian group. Take a presentation of $G$ i.e. a short exact sequence

$$0\rightarrow H \rightarrow F\rightarrow G\rightarrow 0$$

where $F$ a free abelian group. Since $H=ker(F\rightarrow G)$, it is a free abelian group as well and we choose bases $\{f_\alpha\}_\alpha$ for $F$ and $\{h_\beta\}_\beta$ for $H$. We then construct a CW complex

$$X=M(G,n)$$

by taking the $n$-th skeleton to be $X^n:=\vee_\alpha S^n_\alpha$ and for each $\beta$ we attach an $n+1$-cell as follows:

Write $h_\beta=\Sigma_\alpha d_{\alpha\beta}f_\alpha$ and let $\delta_{d_{\alpha\beta}}$ be $0$ if $d_{\alpha\beta}= 0$ and be $1$ otherwise. Define an attaching map $S^n_\beta\rightarrow X^n$ by contracting $\ell_\beta:=(\Sigma_\alpha \delta_{d_{\alpha\beta}})-1$ $(n-1)-$spheres in $S^n$ thus defining a map $S^n_\beta\rightarrow \vee_{\ell_\beta} S^n_{\alpha\beta}$ and then map each $S^n_{\alpha\beta}$ to $S^n_{\alpha}$ by a degree $d_{\alpha\beta}$.

Definition

The (homotopy type of) the topological space $M(G,n)$ constructed this way we call the Moore space of $G$ in degree $n$.

The resulting CW-complex can be seen to have the desired properties via cellular homology.

Properties

Homotopy type

The homotopy type of $M(G,n)$ is determined by specifying $G$ and $n$.

(Non-)Functoriality of the construction

The construction above is not functorial in $G$ because of the choice of bases (see more below). However, it does give a functor to the homotopy category $M(-,n):Ab\rightarrow Ho(Top)$.

The functoriality problem of the construction above cannot be corrected. That is, there is no functor $Ab\rightarrow Top$ that lifts $M(-,n)$. This can be seen as a corollary of a counterexample of Carlsson which gives a negative answer to a conjecture of Steenrod:

Conjecture

(Steenrod)

Given a group $G$ a $G$-module $M$ and a natural number $n$, there is a $G$-space $X$ which has only one non-zero reduced homology G-module in dimension $n$ that satisfy $\tilde{H}_n(X;\mathbb{Z}) \cong M$ as $G$-modules.

Carlsson provides counter examples for such “equivariant Moore spaces” for all non-cyclic groups.

Corollary

There is thus no functor Ab$\rightarrow$ Top that lifts $M(-,n)\colon Ab\rightarrow Ho(Top)$ since if there was such, it would induce, for any group $G$ a functor $Ab^G\rightarrow Top^G$ and in particular a positive answer to the Steenrod conjecture.

Moreover, there can also not be an (∞,1)-functor $Ab\rightarrow L_{whe} Top$ that lifts $M(-,n)$ since this will similarly yield an $\infty$-functor $Ab^G\rightarrow Top^{hG}$ where $Top^{hG}$ is the (∞,1)-category of ∞-actions of $G$ on spaces. Since there is a “rigidification” functor $Top^{hG}\rightarrow Top^G$ this would yield an (ordinary) functor $Ab^G\rightarrow Top^G$ which does not exist by our previous observation.

Related concepts

Co-Moore spaces

There is also a cohomology analogue known as a co-Moore space or a Peterson 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.

Moore decomposition

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.

Related concepts

• Peterson space

• Eilenberg–MacLane space

References

• Example 2.40 of Hatcher Algebraic Topology.

• Marek Golasinski and Daciberg Lima 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.