Moore space
A topological space with a unique non-trivial reduced homology group:
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If is the Eilenberg–MacLane space of the group of integers
and
is the Moore space with
, then
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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
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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
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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) |
Moore space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moore_space&oldid=18047