nLab
Moore space

Contents

Idea

The Moore space M(G,n)M(G, n), where GG is an abelian group and n1n \geq 1, is a topological space which has non-trivial (reduced) homology group GG precisely only in dimension nn.

(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 GG be an abelian group. Take a presentation of GG i.e. a short exact sequence

0HFG0 0\rightarrow H \rightarrow F\rightarrow G\rightarrow 0

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

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

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

Write h β=Σ αd αβf αh_\beta=\Sigma_\alpha d_{\alpha\beta}f_\alpha and let δ d αβ\delta_{d_{\alpha\beta}} be 00 if d αβ=0d_{\alpha\beta}= 0 and d αβ=1d_{\alpha\beta}=1 otherwise. Define an attaching map S β nX nS^n_\beta\rightarrow X^n by contracting β:=(Σ αδ d αβ)1\ell_\beta:=(\Sigma_\alpha \delta_{d_{\alpha\beta}})-1 (n1)(n-1)-spheres in S nS^n thus defining a map S β n βS αβ nS^n_\beta\rightarrow \vee_{\ell_\beta} S^n_{\alpha\beta} and then map each S αβ nS^n_{\alpha\beta} to S α nS^n_{\alpha} by a degree d αβd_{\alpha\beta}.

Definition

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

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)M(G,n) is determined by specifying GG and nn.

(Non-)Functoriality of the construction

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

The functoriality problem of the construction above cannot be corrected. That is, there is no functor AbTopAb\rightarrow Top that lifts M(,n)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 GG a GG-module MM and a natural number nn, there is a GG-space XX which has only one non-zero reduced homology G-module in dimension nn that satisfy H˜ n(X;)M\tilde{H}_n(X;\mathbb{Z}) \cong M as GG-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):AbHo(Top)M(-,n)\colon Ab\rightarrow Ho(Top) since if there was such, it would induce, for any group GG a functor Ab GTop GAb^G\rightarrow Top^G and in particular a positive answer to the Steenrod conjecture.

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

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 GG. Spheres are both Moore and co-Moore spaces for G=G = \mathbb{Z}.

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

According to Baues, Moore spaces are HπH \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.

References

  • 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.

Revised on February 6, 2014 17:06:26 by Beren Sanders (23.243.27.68)