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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Moore space is the name given to a particular type of
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
that is the
homology analogue of the
Eilenberg–Maclane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders 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 mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
, in the sense that it has only one nonzero homology (rather than homotopy) group.
The study of Moore spaces was initiated by
John Coleman Moore
John Coleman Moore (May 27, 1923 – January 1, 2016) was an American mathematician. The Borel−Moore homology and Eilenberg–Moore spectral sequence are named after him.
Early life and education
Moore was born in 1923 in Staten Island, Ne ...
in 1954.
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 commu ...
''G'' and an
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''n'' ≥ 1, let ''X'' be a
CW complex
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
such that
:
and
:
for ''i'' ≠ ''n'', where
denotes the ''n''-th
singular homology group of ''X'' and
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. It's also sensible to require (as Moore did) that ''X'' be simply-connected if ''n''>1.
[Moore 1954]
Examples
*
is a Moore space of
for
.
*
is a Moore space of
for
.
See also
*
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders 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
In algebraic topology, a homology sphere is an ''n''-manifold ''X'' having the homology groups of an ''n''-sphere, for some integer n\ge 1. That is,
:H_0(X,\Z) = H_n(X,\Z) = \Z
and
:H_i(X,\Z) = \ for all other ''i''.
Therefore ''X'' is a conne ...
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
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