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:''See also [[Moore space]] for other meanings in mathematics.'' | |||
In [[algebraic topology]], a branch of [[mathematics]], '''Moore space''' is the name given to a particular type of [[topological space]] that is the [[homology theory|homology]] analogue of the [[Eilenberg–Maclane space]]s of [[homotopy theory]]. | |||
==Formal definition== | |||
Given an [[abelian group]] ''G'' and an [[integer]] ''n'' ≥ 1, let ''X'' be a [[CW complex]] such that | |||
:<math>H_n(X) \cong G</math> | |||
and | |||
:<math>\tilde{H}_i(X) \cong 0</math> | |||
for ''i'' ≠ ''n'', where ''H''<sub>''n''</sub>(''X'') denotes the ''n''-th [[Singular homology|singular homology group]] of ''X'' and <math>\tilde{H}_i(X)</math> is the ''i''th [[reduced homology]] group. Then ''X'' is said to be a '''Moore space'''. | |||
==Examples== | |||
*<math>S^n</math> is a Moore space of <math>\mathbb{Z}</math> for <math>n\geq 1</math>. | |||
*<math>\mathbb{RP}^2</math> is a Moore space of <math>\mathbb{Z}/2\mathbb{Z}</math> (n=1). | |||
==References== | |||
*[[Allen Hatcher|Hatcher, Allen]]. ''Algebraic topology'', Cambridge University Press (2002), ISBN 0-521-79540-0. For further discussion of Moore spaces, see Chapter 2, Example 2.40. A free electronic version of this book is available on the [http://www.math.cornell.edu/~hatcher/ author's homepage]. | |||
[[Category:Algebraic topology]] | |||
{{topology-stub}} | |||
Revision as of 11:58, 28 February 2013
- See also Moore space for other meanings in mathematics.
In algebraic topology, a branch of mathematics, Moore space is the name given to a particular type of topological space that is the homology analogue of the Eilenberg–Maclane spaces of homotopy theory.
Formal definition
Given an abelian group G and an integer n ≥ 1, let X be a CW complex such that
and
for i ≠ n, where Hn(X) denotes the n-th singular homology group of X and is the ith reduced homology group. Then X is said to be a Moore space.
Examples
References
- Hatcher, Allen. Algebraic topology, Cambridge University Press (2002), ISBN 0-521-79540-0. For further discussion of Moore spaces, see Chapter 2, Example 2.40. A free electronic version of this book is available on the author's homepage.