Moore space (algebraic topology)
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In
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, a branch of mathematics, Moore space is the name given to a particular type of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that is the homology analogue of the
Eilenberg–Maclane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
s of homotopy theory, in the sense that it has only one nonzero homology (rather than homotopy) group.


Formal definition

Given an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
''G'' and an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''n'' ≥ 1, let ''X'' be a
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
such that :H_n(X) \cong G and :\tilde_i(X) \cong 0 for ''i'' ≠ ''n'', where H_n(X) denotes the ''n''-th singular homology group of ''X'' and \tilde_i(X) is the ''i''-th
reduced homology In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise stat ...
group. Then ''X'' is said to be a Moore space. Also, ''X'' is by definition simply-connected if ''n''>1.


Examples

*S^n is a Moore space of \mathbb for n\geq 1. *\mathbb^2 is a Moore space of \mathbb/2\mathbb for n=1.


See also

*
Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
, the homotopy analog. * Homology sphere


References

* Hatcher, Allen. ''Algebraic topology'', Cambridge University Press (2002), . For further discussion of Moore spaces, see Chapter 2, Example 2.40. A free electronic version of this book is available on th
author's homepage
Algebraic topology {{topology-stub