# Moore space

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## Homology

A topological space $M$ with a unique non-trivial reduced homology group:

$$\widetilde{H} _ {k} ( M) = G ; \ \ \widetilde{H} _ {i} ( M) = 0 ,\ \ i \neq k .$$

If $K ( \mathbf Z , n )$ is the Eilenberg–MacLane space of the group of integers $\mathbf Z$ and $M _ {k} ( G)$ is the Moore space with $\widetilde{H} _ {k} ( M _ {k} ( G) ) = G$, then

$$\lim\limits _ {N \rightarrow \infty } \ \left [ \Sigma ^ {N+} k X , K ( \mathbf Z , N + n ) \wedge M _ {k} ( G) \right ] \cong H ^ {n} ( X , G ) ,$$

that is, $\{ K ( \mathbf Z , n ) \wedge M _ {k} ( G) \}$ is the spectrum of the cohomology theory $H ^ {*} ( , G )$. This allows one to extend the idea of cohomology with arbitrary coefficients to a generalized cohomology theory. For any spectrum $E$, the spectrum $E \wedge M _ {k} ( G)$ defines a cohomology theory $( E \wedge M _ {k} ( G) ) ^ {*}$, called the $E ^ {*}$- cohomology theory with coefficient group $G$. For the definition of generalized homology theories with coefficients in a group $G$, the so-called co-Moore space $M ^ {k} ( G)$ is used, which is characterized by

$$\widetilde{H} {} ^ {k} ( M ^ {k} ( G) ) = G ,\ \ \widetilde{H} {} ^ {i} ( M ^ {k} ( G) ) = 0 ,\ \ i \neq k .$$

For example, the group $\pi _ {i} ( X , G ) = [ M ^ {k} ( G) , X ]$ is called the homotopy group of the space $X$ with coefficients in $G$. However, the space $M ^ {k} ( G)$ does not exist for all pairs $( G , k )$. If $G$ is a finitely-generated group, then $M ^ {k} ( G)$ does exist.

#### References

 [1] J.C. Moore, "On homotopy groups of spaces with a single non-vanishing homotopy group" Ann. of Math. , 59 : 3 (1954) pp. 549–557

For a construction of a Moore space as a CW-complex with one zero cell and further only cells in dimensions $n$ and $n + 1$, cf. [a1]. The Eilenberg–MacLane space $K ( G , n )$ can be obtained from the Moore space $M ( G , n )$ by killing the higher homotopy groups.

#### References

 [a1] B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. §17

## General topology

In general topology, a Moore space is a regular space with a development: a sequence $\{ {\mathcal U} _ {n} \} _ {n}$ of open coverings such that for every $x$ and every open set $O$ containing $x$ there is an $n$ such that $$\mathop{\rm St} ( x , {\mathcal U} _ {n} ) = \cup \{ {U \in {\mathcal U} _ {n} } : {x \in U } \} \subseteq O ;$$ (in other words, $\{ \mathop{\rm St} ( x , {\mathcal U} _ {x} ) \} _ {n}$ is a neighbourhood base at $x$.)

The idea of a development can be found in [a4] (Axiom 1). Moore spaces are generalizations of metric spaces and one can show that collectionwise normal Moore spaces are metrizable [a2]. The question whether every normal Moore space is metrizable generated lots of research; its solution is described in [a3].

#### References

 [a2] R.H. Bing, "Metrization of topological spaces" Canad. J. Math. , 3 (1951) pp. 175–186 [a3] W.G. Fleissner, "The normal Moore space conjecture and large cardinals" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 733–760 [a4] R.L. Moore, "Foundations of point set theory" , Amer. Math. Soc. (1962)
How to Cite This Entry:
Moore space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moore_space&oldid=51474
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article