Cohomologiaa

Cohomologiaa appears to be a nonstandard spelling of cohomology, a central concept in algebraic topology and geometry that assigns algebraic invariants to topological spaces to distinguish their properties.

Cohomology groups are defined from a cochain complex with coboundary maps, and the n-th cohomology group H^n(X;G) is the quotient of cocycles by coboundaries. Cohomology is contravariant: continuous maps f: X → Y induce homomorphisms f^*: H^n(Y;G) → H^n(X;G). The theory can be developed with various coefficient groups G, often abelian.

Several concrete theories are used in practice. Singular cohomology uses singular simplices to build the cochain complex; de Rham cohomology uses differential forms on smooth manifolds with the exterior derivative; and Čech cohomology uses open covers. Relative cohomology H^n(X,A;G) measures the cohomology of a pair (X,A). Reduced cohomology adjusts H^0 to handle disconnected spaces in a uniform way.

A key feature of cohomology is the cup product, which endows H^*(X;G) with a graded ring structure. The Universal Coefficient Theorem relates cohomology to homology and coefficient groups, while results like Kunneth and Poincaré duality connect cohomology across dimensions and across products or manifolds.

Examples help illustrate the concepts. For a connected space X, H^0(X;G) ≅ G. For the circle S^1 with integer coefficients, H^0(S^1) ≅ Z and H^1(S^1) ≅ Z. Cohomology has wide applications in distinguishing spaces, obstructing constructions, and providing invariants in geometry, topology, and algebraic geometry via sheaf-theoretic methods.