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2. Algebraic topology - Wikipedia

   en.wikipedia.org/wiki/Algebraic_topology

   Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism , though usually most classify up to homotopy equivalence .

3. Künneth theorem - Wikipedia

   en.wikipedia.org/wiki/Künneth_theorem

   In the simplest possible case the relationship is that of a tensor product, but for applications it is very often necessary to apply certain tools of homological algebra to express the answer. A Künneth theorem or Künneth formula is true in many different homology and cohomology theories, and the name has become generic.

4. Torsor (algebraic geometry) - Wikipedia

   en.wikipedia.org/wiki/Torsor_(algebraic_geometry)

   Let be a Grothendieck topology and a scheme.Moreover let be a group scheme over , a -torsor (or principal -bundle) over for the topology (or simply a -torsor when the topology is clear from the context) is the data of a scheme and a morphism : with a -invariant (right) action on that is locally trivial in i.e. there exists a covering {} such that the base change over is isomorphic to the ...

5. Topological algebra - Wikipedia

   en.wikipedia.org/wiki/Topological_algebra

   A topological algebra over a topological field is a topological vector space together with a bilinear multiplication ⋅ : A × A → A {\displaystyle \cdot :A\times A\to A} , ( a , b ) ↦ a ⋅ b {\displaystyle (a,b)\mapsto a\cdot b}

6. Algebraic topology (object) - Wikipedia

   en.wikipedia.org/wiki/Algebraic_topology_(object)

   This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C). Another topology, the geometric topology (also called the Chabauty topology ), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are ...

7. List of algebraic topology topics - Wikipedia

   en.wikipedia.org/wiki/List_of_algebraic_topology...

   Chain (algebraic topology) Betti number; Euler characteristic. Genus; Riemann–Hurwitz formula; Singular homology; Cellular homology; Relative homology; Mayer–Vietoris sequence; Excision theorem; Universal coefficient theorem; Cohomology. List of cohomology theories; Cocycle class; Cup product; Cohomology ring; De Rham cohomology; Čech ...

8. Directed algebraic topology - Wikipedia

   en.wikipedia.org/wiki/Directed_algebraic_topology

   In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets .

9. Homology (mathematics) - Wikipedia

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

   The Betti numbers of the manifold are the rank of the free part of the homology group, and in the special case of surfaces, the torsion part of the homology group only occurs for non-orientable cycles. The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology". [26]