Moore space

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

If is the Eilenberg–MacLane space of the group of integers and is the Moore space with , then

that is, is the spectrum of the cohomology theory . This allows one to extend the idea of cohomology with arbitrary coefficients to a generalized cohomology theory. For any spectrum , the spectrum defines a cohomology theory , called the -cohomology theory with coefficient group . For the definition of generalized homology theories with coefficients in a group , the so-called co-Moore space is used, which is characterized by

For example, the group is called the homotopy group of the space with coefficients in . However, the space does not exist for all pairs . If is a finitely-generated group, then 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 and , cf. [a1]. The Eilenberg–MacLane space can be obtained from the Moore space by killing the higher homotopy groups.

In general topology, a Moore space is a regular space with a development. (A development is a sequence of open coverings such that for every and every open set containing there is an such that

(in other words, is a neighbourhood base at .)

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)