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 simplicesthat is, finite simplicial complexes. 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 the domain, 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 timethe first decade of the twentieth centuryon 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.

Statement

The basic form of the theorem is the following:

In short, the theorem says any continuous map between simplicial complexes is the geometric realization of a simplicial mapping up to homotopy and subdivision.

is called a simplicial approximation.

.

, there exists a natural number, there exists a simplicial approximation

denotes the barycentric subdivision times. In fact, the proof of the theorem shows there is a real number.

. So, we may consider the simplicial approximation theorem as a piecewise linear analog of Whitney approximation theorem.

Applications

Here are some typical applications.

cannot map onto an n is homotpic to a constant map. (Strictly speaking, here the based version of the simplicial approximation theorem is used.)

Here is another more substantial but typical application.

1. (Lefschetz fixed point theorem) admits a fixed point.

)

which is a finite number since X has finite dimension.

Sketch of proof: does not have a fixed point and shall show the trace of it is zero.

Step 1: reduces to the case when X.

Indeed, X here is known to be an ENR, a Euclidean neighborhood retract.

Step 2

This is a linear algebra calculation.

Step 3.)

; note this is not about simplicial approximation.

belong to the same closed simplex; thus, by the early assumption,

It follows:

,

Final step.

by the homotopy invariance of homology.

Proof

Source:

The idea of the proof is quite intuitive; if there are sufficiently many and sufficiently randomly located vertices, then on each simplex, a continuous map can be approximated by a piecewise-linear map; thus globally so.

denote the open star be the Lebesgue number is contained in some open set in the cover.

or

’s as vertices, possibly with repetition.

, we have:

be a convex combination and so

. Then we have

Here, we shall give some rigorous argument on how to find a refinement in which the diameter of each simplex is arbitrarily small (of course, this is intuitively entirely obvious.) The argument is by estimating the diameter of a simplex in a barycentric subdivision as follows.

, we can make the maximal diameter of a simplex arbitrary small by iterating barycentric subdivisions.

Proof: for a chain of faces

is

, we have:

is in

,

is bounded by the claimed bound.

,

References

Further reading

• Raphaël Tinarrage, Simplicial approximation to CW complexes in practice, https://arxiv.org/pdf/2112.07573