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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. Homotopical connectivity - Wikipedia

   en.wikipedia.org/wiki/Homotopical_connectivity

   The two definitions are equivalent. The requirement for an n-connected space consists of requirements for all d ≤ n: The requirement for d=-1 means that X should be nonempty. The requirement for d=0 means that X should be path-connected. The requirement for any d ≥ 1 means that X contains no holes of boundary dimension d.

8. Glossary of algebraic topology - Wikipedia

   en.wikipedia.org/wiki/Glossary_of_algebraic_topology

   This is a glossary of properties and concepts in algebraic topology in mathematics. See also: glossary of topology, list of algebraic topology topics, glossary of category theory, glossary of differential geometry and topology, Timeline of manifolds. Convention: Throughout the article, I denotes the unit interval, S n the n-sphere and D n the n ...

9. 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 ...