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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
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 ...
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).
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
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]
Category theory is the language of modern algebra, and has been widely used in the study of algebraic geometry and topology. It has been noted that "the key observation of [10] is that the persistence diagram produced by [8] depends only on the algebraic structure carried by this diagram."
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,*)} .
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 ...