Cohomology ring

In

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

${\displaystyle H^{k}(X;R)\times H^{\ell }(X;R)\to H^{k+\ell }(X;R).}$

The cup product gives a multiplication on the direct sum of the cohomology groups

${\displaystyle H^{\bullet }(X;R)=\bigoplus _{k\in \mathbb {N} }H^{k}(X;R).}$

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

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

${\displaystyle (\alpha ^{k}\smile \beta ^{\ell })=(-1)^{k\ell }(\beta ^{\ell }\smile \alpha ^{k}).}$

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.

Examples

• ${\displaystyle \operatorname {H} ^{*}(\mathbb {R} P^{n};\mathbb {F} _{2})=\mathbb {F} _{2}[\alpha ]/(\alpha ^{n+1})}$ where ${\displaystyle |\alpha |=1}$.
• ${\displaystyle \operatorname {H} ^{*}(\mathbb {R} P^{\infty };\mathbb {F} _{2})=\mathbb {F} _{2}[\alpha ]}$ where ${\displaystyle |\alpha |=1}$.
• ${\displaystyle \operatorname {H} ^{*}(\mathbb {C} P^{n};\mathbb {Z} )=\mathbb {Z} [\alpha ]/(\alpha ^{n+1})}$ where ${\displaystyle |\alpha |=2}$.
• ${\displaystyle \operatorname {H} ^{*}(\mathbb {C} P^{\infty };\mathbb {Z} )=\mathbb {Z} [\alpha ]}$ where ${\displaystyle |\alpha |=2}$.
• ${\displaystyle \operatorname {H} ^{*}(\mathbb {H} P^{n};\mathbb {Z} )=\mathbb {Z} [\alpha ]/(\alpha ^{n+1})}$ where ${\displaystyle |\alpha |=4}$.
• ${\displaystyle \operatorname {H} ^{*}(\mathbb {H} P^{\infty };\mathbb {Z} )=\mathbb {Z} [\alpha ]}$ where ${\displaystyle |\alpha |=4}$.
• By the
  Künneth formula
  , the mod 2 cohomology ring of the cartesian product of n copies of ${\displaystyle \mathbb {R} P^{\infty }}$ is a polynomial ring in n variables with coefficients in ${\displaystyle \mathbb {F} _{2}}$.
• 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.

See also

• Quantum cohomology

References

• Novikov, S. P. (1996). Topology I, General Survey. Springer-Verlag.
  ISBN 7-03-016673-6
  .
• Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press,
  ISBN 0-521-79540-0
  .