Moore space (algebraic topology) explained
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
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.H_1=0 is not enough to guarantee that it's simply connected. See talk page for more.. November 2023.
Examples
is a Moore space of
for
.
is a Moore space of
for
.
See also
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 the author's homepage.