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  2. Cohomology - Wikipedia

    en.wikipedia.org/wiki/Cohomology

    If one prefers homology or cohomology theories to be defined on all topological spaces rather than on CW complexes, one standard approach is to include the axiom that every weak homotopy equivalence induces an isomorphism on homology or cohomology. (That is true for singular homology or singular cohomology, but not for sheaf cohomology, for ...

  3. Homology (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Homology_(mathematics)

    The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups (which are called cohomology groups in this context and denoted by H n) form contravariant functors from the category that X belongs to into the category of abelian groups or ...

  4. Group cohomology - Wikipedia

    en.wikipedia.org/wiki/Group_cohomology

    Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups. The group homology H ∗ ( G , k ) {\displaystyle H_{*}(G,k)} of abelian groups G with values in a principal ideal domain k is closely related to the exterior algebra ∧ ∗ ( G ⊗ ...

  5. Chain complex - Wikipedia

    en.wikipedia.org/wiki/Chain_complex

    The homology of a cochain complex is called its cohomology. In algebraic topology , the singular chain complex of a topological space X is constructed using continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex.

  6. Universal coefficient theorem - Wikipedia

    en.wikipedia.org/wiki/Universal_coefficient_theorem

    An alternative point of view can be based on representing cohomology via Eilenberg–MacLane space, where the map takes a homotopy class of maps (,) to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor. [1]

  7. Relative homology - Wikipedia

    en.wikipedia.org/wiki/Relative_homology

    In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.

  8. Homological algebra - Wikipedia

    en.wikipedia.org/wiki/Homological_algebra

    A central concept is that of chain complexes, which can be studied through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings , modules, topological spaces , and other "tangible" mathematical objects.

  9. List of cohomology theories - Wikipedia

    en.wikipedia.org/wiki/List_of_cohomology_theories

    MO is a rather weak cobordism theory, as the spectrum MO is isomorphic to H(π * (MO)) ("homology with coefficients in π * (MO)") – MO is a product of Eilenberg–MacLane spectra. In other words, the corresponding homology and cohomology theories are no more powerful than homology and cohomology with coefficients in Z/2Z. This was the first ...