Moore space
Encyclopedia
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.
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.
Algebraic topology
Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although algebraic topology...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Moore space is the name given to a particular type of topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
that is the homology
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
analogue of the Eilenberg–Maclane spaces of homotopy theory.
Formal definition
Given an abelian groupAbelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
G and an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
n ≥ 1, let X be a CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
such that
and
for i ≠ n, where Hn(X) denotes the n-th singular homology group
Singular homology
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
of X and
is the ith reduced homologyReduced homology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, designed to make a point have all its homology groups zero...
group. Then X is said to be a Moore space.
Examples
is a Moore space of 
for 
.
is a Moore space of 
(n=1).