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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 ...
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 .
A torus, one of the most frequently studied objects in algebraic topology. 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.
In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure.. Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. [1]
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 ...
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces The main article for this category is Algebraic topology . Contents
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz , and generalizes earlier results of Henri Poincaré .
In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms.Given a topological space and subspaces and such that is also a subspace of , the theorem says that under certain circumstances, we can cut out (excise) from both spaces such that the relative homologies of the pairs (,) into (,) are isomorphic.
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