Moore space

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

Comments

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.

In general topology, a Moore space is a regular space with a development. (A development is 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

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