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  2. Cap product - Wikipedia

    en.wikipedia.org/wiki/Cap_product

    In analogy with the interpretation of the cup product in terms of the Künneth formula, we can explain the existence of the cap product in the following way.Using CW approximation we may assume that is a CW-complex and () (and ()) is the complex of its cellular chains (or cochains, respectively).

  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. Although algebraic topology primarily uses algebra to study topological ...

  4. 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 ...

  5. Fibration - Wikipedia

    en.wikipedia.org/wiki/Fibration

    A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406

  6. Arithmetic topology - Wikipedia

    en.wikipedia.org/wiki/Arithmetic_topology

    Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds . Analogies

  7. Products in algebraic topology - Wikipedia

    en.wikipedia.org/wiki/Products_in_algebraic_topology

    Differential graded algebra: the algebraic structure arising on the cochain level for the cup product; Poincaré duality: swaps some of these; Intersection theory: for a similar theory in algebraic geometry

  8. Cup product - Wikipedia

    en.wikipedia.org/wiki/Cup_product

    In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q.This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H ∗ (X), called the cohomology ring.

  9. Path space (algebraic topology) - Wikipedia

    en.wikipedia.org/.../Path_space_(algebraic_topology)

    In algebraic topology, a branch of mathematics, the based path space of a pointed space (,) is the space that consists of all maps from the interval = [,] to X such that () =, called based paths. [1] In other words, it is the mapping space from ( I , 0 ) {\displaystyle (I,0)} to ( X , ∗ ) {\displaystyle (X,*)} .

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