Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a way of attaching algebraic invariants to a topological space or other mathematical object that encode its properties in a way that is often computable. Cohomology is often related to questions of whether some local property of the space is obstructed when passing to a global property. For example, the Möbius strip is not the product space of a line segment with a circle (i.e., a cylinder), but locally on any segment of the circle it resembles an ordinary rectangle (product of two line segments) without the global twist. The obstruction to making the product structure global is encoded in the first cohomology of the underlying circle, which classifies the two inequivalent ways a line can twist around a circle (an even number of twists or an odd number of twists). The fact that there are only two inequivalent ways of twisting a line around a circle is encoded in the relevant cohomology group ${\displaystyle H^{1}(S^{1},\{+1,-1\})}$ which can be computed as isomorphic to the group ${\displaystyle \{+1,-1\}}$ where ${\displaystyle +1}$ corresponds to an even number of twists, and ${\displaystyle -1}$ to an odd number.

In its general abstract formulation, cohomology is a sequence of abelian groups often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology.

From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with functions and pullbacks in geometric situations: given spaces ${\displaystyle X}$ and ${\displaystyle Y}$, and some function ${\displaystyle F}$ on ${\displaystyle Y}$, for any mapping ${\displaystyle f:X\to Y}$, composition with ${\displaystyle F}$ gives rise to a function ${\displaystyle F\circ f}$ on ${\displaystyle X}$. The most important cohomology theories have a product, the cup product, which gives them a ring structure. Because of this feature, cohomology is usually a stronger invariant than homology.