# Moore space

## 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.

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