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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 ...
Let be a Grothendieck topology and a scheme.Moreover let be a group scheme over , a -torsor (or principal -bundle) over for the topology (or simply a -torsor when the topology is clear from the context) is the data of a scheme and a morphism : with a -invariant (right) action on that is locally trivial in i.e. there exists a covering {} such that the base change over is isomorphic to the ...
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 ...
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology.
In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets .
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces Subcategories. This category has the following ...
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 ...
A central result of the theory is that the rational homotopy category can be described in a purely algebraic way; in fact, in two different algebraic ways. First, Quillen showed that the rational homotopy category is equivalent to the homotopy category of connected differential graded Lie algebras.