From Wikipedia, the free encyclopedia
In mathematics , specifically algebraic topology , the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology , but the ring structure is also present in other theories such as de Rham cohomology . It is also functorial : for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant.
Specifically, given a sequence of cohomology groups H k X ;R ) on X with coefficients in a commutative ring R (typically R is Z n Z , Q , R , or C ) one can define the cup product , which takes the form
H
k
(
X
;
R
)
×
H
ℓ
(
X
;
R
)
→
H
k
+
ℓ
(
X
;
R
)
.
{\displaystyle H^{k}(X;R)\times H^{\ell }(X;R)\to H^{k+\ell }(X;R).}
The cup product gives a multiplication on the direct sum of the cohomology groups
H
∙
(
X
;
R
)
=
⨁
k
∈
N
H
k
(
X
;
R
)
.
{\displaystyle H^{\bullet }(X;R)=\bigoplus _{k\in \mathbb {N} }H^{k}(X;R).}
This multiplication turns H • (X ;R ) into a ring. In fact, it is naturally an N -graded ring with the nonnegative integer k serving as the degree. The cup product respects this grading.
The cohomology ring is graded-commutative in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree k and ℓ; we have
(
α
k
⌣
β
ℓ
)
=
(
−
1
)
k
ℓ
(
β
ℓ
⌣
α
k
)
.
{\displaystyle (\alpha ^{k}\smile \beta ^{\ell })=(-1)^{k\ell }(\beta ^{\ell }\smile \alpha ^{k}).}
A numerical invariant derived from the cohomology ring is the cup-length , which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example, a complex projective space has cup-length equal to its complex dimension .
H
∗
(
R
P
n
;
F
2
)
=
F
2
[
α
]
/
(
α
n
+
1
)
{\displaystyle \operatorname {H} ^{*}(\mathbb {R} P^{n};\mathbb {F} _{2})=\mathbb {F} _{2}[\alpha ]/(\alpha ^{n+1})}
|
α
|
=
1
{\displaystyle |\alpha |=1}
H
∗
(
R
P
∞
;
F
2
)
=
F
2
[
α
]
{\displaystyle \operatorname {H} ^{*}(\mathbb {R} P^{\infty };\mathbb {F} _{2})=\mathbb {F} _{2}[\alpha ]}
|
α
|
=
1
{\displaystyle |\alpha |=1}
H
∗
(
C
P
n
;
Z
)
=
Z
[
α
]
/
(
α
n
+
1
)
{\displaystyle \operatorname {H} ^{*}(\mathbb {C} P^{n};\mathbb {Z} )=\mathbb {Z} [\alpha ]/(\alpha ^{n+1})}
|
α
|
=
2
{\displaystyle |\alpha |=2}
H
∗
(
C
P
∞
;
Z
)
=
Z
[
α
]
{\displaystyle \operatorname {H} ^{*}(\mathbb {C} P^{\infty };\mathbb {Z} )=\mathbb {Z} [\alpha ]}
|
α
|
=
2
{\displaystyle |\alpha |=2}
H
∗
(
H
P
n
;
Z
)
=
Z
[
α
]
/
(
α
n
+
1
)
{\displaystyle \operatorname {H} ^{*}(\mathbb {H} P^{n};\mathbb {Z} )=\mathbb {Z} [\alpha ]/(\alpha ^{n+1})}
|
α
|
=
4
{\displaystyle |\alpha |=4}
H
∗
(
H
P
∞
;
Z
)
=
Z
[
α
]
{\displaystyle \operatorname {H} ^{*}(\mathbb {H} P^{\infty };\mathbb {Z} )=\mathbb {Z} [\alpha ]}
|
α
|
=
4
{\displaystyle |\alpha |=4}
H
∗
(
T
2
;
Z
)
=
Λ
Z
[
α
1
,
α
2
]
{\displaystyle \operatorname {H} ^{*}(T^{2};\mathbb {Z} )=\Lambda _{\mathbb {Z} }[\alpha _{1},\alpha _{2}]}
|
α
1
|
=
|
α
2
|
=
1
{\displaystyle |\alpha _{1}|=|\alpha _{2}|=1}
H
∗
(
T
n
;
Z
)
=
Λ
Z
[
α
1
,
.
.
.
,
α
n
]
{\displaystyle \operatorname {H} ^{*}(T^{n};\mathbb {Z} )=\Lambda _{\mathbb {Z} }[\alpha _{1},...,\alpha _{n}]}
|
α
i
|
=
1
{\displaystyle |\alpha _{i}|=1}
H
∗
(
S
n
;
Z
)
=
Z
[
α
]
/
[
α
2
]
{\displaystyle \operatorname {H} ^{*}(S^{n};\mathbb {Z} )=\mathbb {Z} [\alpha ]/[\alpha ^{2}]}
|
α
|
=
n
{\displaystyle |\alpha |=n}
If
K
{\displaystyle K}
H
∗
(
K
;
Z
)
=
Z
[
α
,
β
]
/
[
α
2
,
2
β
,
α
β
]
{\displaystyle \operatorname {H} ^{*}(K;\mathbb {Z} )=\mathbb {Z} [\alpha ,\beta ]/[\alpha ^{2},2\beta ,\alpha \beta ]}
|
α
|
=
1
,
|
β
|
=
2
{\displaystyle |\alpha |=1,|\beta |=2}
By the Künneth formula , the mod 2 cohomology ring of the cartesian product of n copies of
R
P
∞
{\displaystyle \mathbb {R} P^{\infty }}
n variables with coefficients in
F
2
{\displaystyle \mathbb {F} _{2}}
The reduced cohomology ring of wedge sums is the direct product of their reduced cohomology rings.
The cohomology ring of suspensions vanishes except for the degree 0 part. 
![{\displaystyle \operatorname {H} ^{*}(\mathbb {R} P^{n};\mathbb {F} _{2})=\mathbb {F} _{2}[\alpha ]/(\alpha ^{n+1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b18ec9321d92c60d530b5bd2813aca4dfbeae6)
![{\displaystyle \operatorname {H} ^{*}(\mathbb {R} P^{\infty };\mathbb {F} _{2})=\mathbb {F} _{2}[\alpha ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/770e7d6ddbb2b3a21381e6fe471864586e22852e)
![{\displaystyle \operatorname {H} ^{*}(\mathbb {C} P^{n};\mathbb {Z} )=\mathbb {Z} [\alpha ]/(\alpha ^{n+1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c3c9beff6ca1510d126df1e847980f845c8f99)
![{\displaystyle \operatorname {H} ^{*}(\mathbb {C} P^{\infty };\mathbb {Z} )=\mathbb {Z} [\alpha ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67cf6b8898ad7b7df0f0422a3dd6f746c7c8daa5)
![{\displaystyle \operatorname {H} ^{*}(\mathbb {H} P^{n};\mathbb {Z} )=\mathbb {Z} [\alpha ]/(\alpha ^{n+1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23e588e528737fc89dd0b678a0268bf144659f67)
![{\displaystyle \operatorname {H} ^{*}(\mathbb {H} P^{\infty };\mathbb {Z} )=\mathbb {Z} [\alpha ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/590a3b6b8e1b171a28bea577386cb4c7fed9f968)
![{\displaystyle \operatorname {H} ^{*}(T^{2};\mathbb {Z} )=\Lambda _{\mathbb {Z} }[\alpha _{1},\alpha _{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4aa5d291c9cd11fbc9ad347257c1ba19683068f)
![{\displaystyle \operatorname {H} ^{*}(T^{n};\mathbb {Z} )=\Lambda _{\mathbb {Z} }[\alpha _{1},...,\alpha _{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ffd515c97f3ecf0e0af5011503acd59416fe881)
![{\displaystyle \operatorname {H} ^{*}(S^{n};\mathbb {Z} )=\mathbb {Z} [\alpha ]/[\alpha ^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/225f3d1eca5cd7a5ca945a6584456ec481f98515)
![{\displaystyle \operatorname {H} ^{*}(K;\mathbb {Z} )=\mathbb {Z} [\alpha ,\beta ]/[\alpha ^{2},2\beta ,\alpha \beta ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5337e07b05d3e51138956c6e9de453b81619a12)
