Cohomology ($H^{\ast }(X;R)$)

Cohomology is the contravariant counterpart of homology, formed from cochains and coboundaries. A coefficient ring equips it with the cup product and hence with a graded ring structure.

For an abelian coefficient group $G$, cohomology fits into a split short exact sequence

Compactly supported cohomology is the cohomology of cochains vanishing outside a compact subset. Equivalently,

A continuous map is proper when inverse images of compact sets are compact. Proper maps induce contravariant maps on compactly supported cohomology.

The cap product pairs a homology class and a cohomology class to lower degree:

For an oriented closed connected $d$-manifold, the fundamental class is the unique generator of $H_d(M;\mathbb{Z})$ that restricts to the orientation generator in every local homology group.

For a closed oriented $d$-manifold, cap product with the fundamental class gives isomorphisms

If a closed oriented manifold is covered by two open sets and a positive-degree cohomology class restricts to zero on both, then its cap product with the fundamental class vanishes in every degree strictly below the top. A chain proof uses the small simplex theorem and replaces the two restricted cocycles by coboundaries.

The cup product is a natural bilinear operation $H^p(X;R)\times H^q(X;R)\to H^{p+q}(X;R)$. It is graded-commutative: $a⌣b=(-1{)}^{pq}b⌣a$.

The cohomology ring is the graded ring whose multiplication is the cup product. It records intersection information that the graded cohomology groups alone do not detect.

Choose degree-one classes ${\alpha }_i,{\beta }_i$ dual to a symplectic basis of one-cycles and let $\omega$ generate $H^2({\Sigma }_g;\mathbb{Z})$. The only nonzero products of positive-degree basis elements are

The Künneth theorem computes the homology or cohomology of a product from those of its factors. Over a field $k$, $H^{\ast }(X\times Y;k)\cong H^{\ast }(X;k){\otimes }_k{H}^{\ast }(Y;k)$ as graded rings under suitable finiteness hypotheses.