This has the effect of "reversing all the arrows" of the original complex, leaving a cochain complex

Some of the formal properties of cohomology are only minor variants of the properties of homology:

In what follows, cohomology is taken with coefficients in the integers Z, unless stated otherwise.

In 1944, Samuel Eilenberg overcame the technical limitations, and gave the modern definition of singular homology and cohomology.

Many of these theories carry richer information than ordinary cohomology, but are harder to compute.

Cohomology theories in a broader sense (invariants of other algebraic or geometric structures, rather than of topological spaces) include: