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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 .
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 ...
A topological algebra over a topological field is a topological vector space together with a bilinear multiplication ⋅ : A × A → A {\displaystyle \cdot :A\times A\to A} , ( a , b ) ↦ a ⋅ b {\displaystyle (a,b)\mapsto a\cdot b}
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group , denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space .
An algebraic cycle on X is a formal combination of subvarieties of X; that is, it is something of the form . The coefficients are usually taken to be integral or rational. We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components.
Ronald Brown "Topology and Groupoids" pdf available Gives an account of some categorical methods in topology, use the fundamental groupoid on a set of base points to give a generalisation of the Seifert-van Kampen Theorem. Philip J. Higgins, "Categories and Groupoids" free download Explains some uses of groupoids in group theory and 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 ...
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 ...
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