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  2. Betti number - Wikipedia

    en.wikipedia.org/wiki/Betti_number

    In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they ...

  3. 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 .

  4. Alexander–Spanier cohomology - Wikipedia

    en.wikipedia.org/wiki/Alexander–Spanier_cohomology

    A subset is said to be cobounded if is bounded, i.e. its closure is compact.. Similar to the definition of Alexander cohomology module, one can define Alexander cohomology module with compact supports of a pair (,) by adding the property that (,;) is locally zero on some cobounded subset of .

  5. List of algebraic topology topics - Wikipedia

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

    Path (topology) Fundamental group; Homotopy group; Seifert–van Kampen theorem; Pointed space; Winding number; Simply connected. Universal cover; Monodromy; Homotopy lifting property; Mapping cylinder; Mapping cone (topology) Wedge sum; Smash product; Adjunction space; Cohomotopy; Cohomotopy group; Brown's representability theorem; Eilenberg ...

  6. 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]

  7. Local system - Wikipedia

    en.wikipedia.org/wiki/Local_system

    In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point.

  8. Étale fundamental group - Wikipedia

    en.wikipedia.org/wiki/Étale_fundamental_group

    In algebraic topology, the fundamental group (,) of a pointed topological space (,) is defined as the group of homotopy classes of loops based at .This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology.

  9. Ext functor - Wikipedia

    en.wikipedia.org/wiki/Ext_functor

    Here are some of the basic properties and computations of Ext groups. [3]Ext 0 R (A, B) ≅ Hom R (A, B) for any R-modules A and B.; Ext i R (A, B) = 0 for all i > 0 if the R-module A is projective (for example, free) or if B is injective.

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