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Universal coefficient theorem for homology
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Mathematics
Area of mathematics
Geometry and topology
Algebraic topology
Homology
Created
2026-09-24
Updated
2026-09-24
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For an abelian
coefficient
group
G
, there is
a
split
short exact sequence
0
→
H
i
(
X
;
Z
)
⊗
G
→
H
i
(
X
;
G
)
→
Tor
(
H
i
−
1
(
X
;
Z
)
,
G
)
→
0.
(1)
For
G
=
Q
, the Tor term vanishes.
Table of contents
Rational homology
Universal coefficient theorem for homology
Rational homology
0
0
0
Universal coefficient theorem for homology
Rational homology
satisfies
H
i
(
X
;
Q
)
≅
H
i
(
X
;
Z
)
⊗
Z
Q
.
(1)
It records the
free
rank of
integral
homology
and discards torsion.
Ancestors
(6)
Homology
Algebraic topology
Geometry and topology
Area of mathematics
Mathematics
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