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Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
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.
Fundamental groups also appear in algebraic geometry and are the main topic of Alexander Grothendieck's first Séminaire de géométrie algébrique (SGA1). A version of Van Kampen's theorem appears there, and is proved along quite different lines than in algebraic topology, namely by descent theory. A similar proof works in algebraic topology. [18]
The usual proof of this result is a pure piece of homological algebra about chain complexes of free ... geometrically flavored introduction to algebraic topology.
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties.
The method of proof we shall use is commonly referred to as diagram chasing. [1] We shall prove the five lemma by individually proving each of the two four lemmas. To perform diagram chasing, we assume that we are in a category of modules over some ring , so that we may speak of elements of the objects in the diagram and think of the morphisms ...
The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis. The case n = 3 first was proved by Piers Bohl in 1904 (published in Journal für die reine und angewandte Mathematik ). [ 14 ]