Algebraic Topology
Introduction
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.
Fundamental Concepts
The fundamental group and homology are invariants of algebraic topology. In less formal terms, algebraic topology allows for the algebraic study of topological spaces. The discipline that applies algebraic objects to topology, primarily topological spaces, is known as Algebraic Topology.
Homotopy Groups
The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
Homology and Cohomology
Homology and cohomology groups, on the other hand, are functorial in nature, meaning they are more amenable to computation. They are derived from the chain complex, a sequence of abelian groups or modules linked by homomorphisms.
Homotopy Theory
Homotopy theory is a subfield of algebraic topology that deals with the algebraic invariants of spaces under homotopy equivalence.
Applications
Algebraic topology finds application in various areas of mathematics, including functional analysis, differential geometry, algebraic geometry, and number theory. It also has important applications in theoretical physics, particularly in string theory and condensed matter physics.
History
Algebraic topology, as a well-defined mathematical discipline, originates in the first part of the twentieth century under the influence of Henri Poincaré. The development of algebraic topology spins from the fields of combinatorial topology (also known as Betti numbers, singular homology) and differential topology (also known as de Rham cohomology).