Moore space (algebraic topology): Difference between revisions
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for ''i'' ≠ ''n'', where <math>H_n(X)</math> 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'''. |
for ''i'' ≠ ''n'', where <math>H_n(X)</math> 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'''. Some authors also require that ''X'' be simply-connected if ''n''>1.{{cite-needed|reason=H_1=0 is not enough to guarantee that it's simply connected. See talk page for more.|date=November 2023}} |
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==Examples== |
==Examples== |
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==References== |
==References== |
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*[[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:// |
*[[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://pi.math.cornell.edu/~hatcher/ author's homepage]. |
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[[Category:Algebraic topology]] |
[[Category:Algebraic topology]] |
Latest revision as of 02:12, 28 September 2024
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, in the sense that it has only one nonzero homology (rather than homotopy) group.
Formal definition
[edit]Given an abelian group G and an integer n ≥ 1, let X be a CW complex such that
and
for i ≠ n, where denotes the n-th singular homology group of X and is the i-th reduced homology group. Then X is said to be a Moore space. Some authors also require that X be simply-connected if n>1.[citation needed]
Examples
[edit]- is a Moore space of for .
- is a Moore space of for .
See also
[edit]- Eilenberg–MacLane space, the homotopy analog.
- Homology sphere
References
[edit]- 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.