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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 ...
Tammo tom Dieck 1972. Tammo tom Dieck (29 May 1938, São Paulo) is a German mathematician, specializing in algebraic topology.. Tammo tom Dieck studied mathematics from 1957 at the University of Göttingen and at Saarland University, where he received his promotion (Ph.D.) in 1964 under Dieter Puppe with thesis Zur -Theorie und ihren Kohomologie-Operationen. [1]
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 quantum field theory, there exist quantum categories. [16] [17] [18] and quantum double groupoids. [18]One can consider quantum double groupoids to be fundamental groupoids defined via a 2-functor, which allows one to think about the physically interesting case of quantum fundamental groupoids (QFGs) in terms of the bicategory Span(Groupoids), and then constructing 2-Hilbert spaces and 2 ...
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type. What this looks like is for a space X {\displaystyle X} there is a list of spaces { X n } n ≥ 0 {\displaystyle \{X_{n}\}_{n\geq 0}} where
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
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 ...