Cohomology

Uncommon card · No. 182224

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...

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