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    Cohomology - Wikipedia

    • In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, 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 coh… See more

    Singular cohomology

    Singular cohomology is a powerful invariant in topology, associating a graded-commutative ring with any topological … See more

    Examples

    In what follows, cohomology is taken with coefficients in the integers Z, unless stated otherwise.
    • The cohomology ring of a point is the ring Z in degree 0. By homotopy invariance, this is also the … See more

    Characteristic classes

    An oriented real vector bundle E of rank r over a topological space X determines a cohomology class on X, the Euler class χ(E) ∈ H (X,Z). Informally, the Euler class is the class of the zero set of a general section of E. That inte… See more

    Eilenberg–MacLane spaces

    For each abelian group A and natural number j, there is a space whose j-th homotopy group is isomorphic to A and whose other homotopy groups are zero. Such a space is called an … See more

    Cap product

    For any topological space X, the cap product is a bilinear map
    for any integers i and j and any commutative ring R. The resulting map
    makes the singular homology of X into a module ove… See more

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