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

Hn(X)\congG

and

\tilde{H}i(X)\cong0

for in, where

Hn(X)

denotes the n-th singular homology group of X and

\tilde{H}i(X)

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

Sn

is a Moore space of

Z

for

n\geq1

.

RP2

is a Moore space of

Z/2Z

for

n=1

.

See also

References