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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. 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}

  4. Homotopy group - Wikipedia

    en.wikipedia.org/wiki/Homotopy_group

    In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group , denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space .

  5. Homotopy theory - Wikipedia

    en.wikipedia.org/wiki/Homotopy_theory

    In homotopy theory and algebraic topology, the word "space" denotes a topological space.In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex.

  6. A¹ homotopy theory - Wikipedia

    en.wikipedia.org/wiki/A¹_homotopy_theory

    In algebraic geometry and algebraic topology, branches of mathematics, A 1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky.

  7. Mapping cylinder - Wikipedia

    en.wikipedia.org/wiki/Mapping_cylinder

    In mathematics, specifically algebraic topology, the mapping cylinder [1] of a continuous function between topological spaces and is the quotient = (([,])) / where the denotes the disjoint union, and ~ is the equivalence relation generated by

  8. Mayer–Vietoris sequence - Wikipedia

    en.wikipedia.org/wiki/Mayer–Vietoris_sequence

    Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. [8]

  9. Grothendieck's Galois theory - Wikipedia

    en.wikipedia.org/wiki/Grothendieck's_Galois_theory

    In the above example, a connection with classical Galois theory can be seen by regarding ^ as the profinite Galois group Gal(F /F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F .

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