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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).
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 ...
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 ...
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
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
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
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.
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,*)} .