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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 .
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space [note 1] is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer . A connected topological space X is called an Eilenberg–MacLane space of type K ( G , n ) {\displaystyle K(G,n)} , if it has n -th homotopy group π n ...
The name comes from a relation between the first Tor group Tor 1 and the torsion subgroup of an abelian group. In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950. [1] It was first applied to the Künneth theorem and universal coefficient theorem in topology.
Massey, William S. (1991), A Basic Course in Algebraic Topology, Springer, ISBN 038797430X; May, J. Peter (1999), A Concise Course in Algebraic Topology, ISBN 9780226511832; Deane Montgomery and Leo Zippin, Topological Transformation Groups, Interscience Publishers (1955) Munkres, James R. (2000), Topology, Prentice Hall, ISBN 0-13-181629-2
In algebraic topology, the fundamental group (,) of a pointed topological space (,) is defined as the group of homotopy classes of loops based at . This definition works well for spaces such as real and complex manifolds , but gives undesirable results for an algebraic variety with the Zariski topology .
The degree of a map between general manifolds was first defined by Brouwer, [1] who showed that the degree is homotopy invariant and used it to prove the Brouwer fixed point theorem. Less general forms of the concept existed before Brouwer, such as the winding number and the Kronecker characteristic (or Kronecker integral ).
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}
Call a cohomology theory even periodic if = for i odd and there is an invertible element .These theories possess a complex orientation, which gives a formal group law.A particularly rich source for formal group laws are elliptic curves.