Gluing axiom

The gluing axiom (also sometimes referred to as the Mayer-Vietoris axiom in some contexts, though Mayer-Vietoris more generally refers to a specific computational sequence derived from the gluing property) is a fundamental property in cohomology theories, particularly sheaf cohomology and singular cohomology. It essentially states that if a topological space can be covered by a collection of open sets, then the cohomology of the entire space can be determined from the cohomology of the individual open sets and their intersections, along with some connecting homomorphisms.

More formally, the gluing axiom describes how cohomology classes defined on the intersection of open sets within a covering can be "glued" together to form a cohomology class on the entire space, under certain conditions. It provides a way to build global information from local information, which is a central theme in many areas of mathematics.

In sheaf cohomology, the gluing axiom arises naturally from the definition of a sheaf itself. A sheaf is a collection of data associated to open sets, along with restriction maps. The gluing axiom essentially ensures that this data is consistent, so that if we have sections (elements of the sheaf) defined on overlapping open sets that agree on the overlaps, then we can combine them to form a section defined on the union of the open sets.

The gluing axiom is a key part of the Eilenberg-Steenrod axioms, which axiomatize homology and cohomology theories. Theories satisfying these axioms are considered to be well-behaved and geometrically meaningful. The gluing axiom, along with the other axioms (such as homotopy invariance, excision, and the dimension axiom), allows us to compute the cohomology of complex spaces by breaking them down into simpler pieces.