nLab
Moore space

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

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 be 11 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 October 10, 2015 02:49:57 by Tim Porter (90.24.161.105)