Simplicial approximation theorem
Encyclopedia
In mathematics
, the simplicial approximation theorem is a foundational result for algebraic topology
, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise
of the simplest kind. It applies to mappings between spaces that are built up from simplices
— that is, finite simplicial complex
es. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (affine-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision
of the simplices of both the domain and range, and (ii) replacement of the actual mapping by a homotopic one.
This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness). It served to put the homology theory
of the time — the first decade of the twentieth century — on a rigorous basis, since it showed that the topological effect (on homology groups) of continuous mappings could in a given case be expressed in a finitary
way. This must be seen against the background of a realisation at the time that continuity was in general compatible with the pathological
, in some other areas. This initiated, one could say, the era of combinatorial topology
.
There is a further simplicial approximation theorem for homotopies, stating that a homotopy
between continuous mappings can likewise be approximated by a combinatorial version.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the simplicial approximation theorem is a foundational result for algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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.Although algebraic topology...
, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise
Piecewise
On mathematics, a piecewise-defined function is a function whose definition changes depending on the value of the independent variable...
of the simplest kind. It applies to mappings between spaces that are built up from simplices
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
— that is, finite simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (affine-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision
Barycentric subdivision
In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.The name...
of the simplices of both the domain and range, and (ii) replacement of the actual mapping by a homotopic one.
This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness). It served to put the homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
of the time — the first decade of the twentieth century — on a rigorous basis, since it showed that the topological effect (on homology groups) of continuous mappings could in a given case be expressed in a finitary
Finitary
In mathematics or logic, a finitary operation is one, like those of arithmetic, that takes a finite number of input values to produce an output. An operation such as taking an integral of a function, in calculus, is defined in such a way as to depend on all the values of the function , and is so...
way. This must be seen against the background of a realisation at the time that continuity was in general compatible with the pathological
Pathological (mathematics)
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved....
, in some other areas. This initiated, one could say, the era of combinatorial topology
Combinatorial topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces were regarded as derived from combinatorial decompositions such as simplicial complexes...
.
There is a further simplicial approximation theorem for homotopies, stating that a homotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
between continuous mappings can likewise be approximated by a combinatorial version.