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
editGiven 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.