Simplicial approximation

In mathematics , especially in algebraic topology , the simplicial approximation of a continuous mapping is an important tool for combining combinatorial and continuous methods. The simplicial approximation theorem states that every continuous mapping between simplicial complexes (after a sufficiently fine subdivision) can be approximated by simplicial mappings. It was proven around 1910 by Luitzen Brouwer , who used it to prove the topological invariance of simplicial homology and thus to secure the foundations of the homology theory of that time.

Definition: Simplicial Approximation

Given are simplicial complexes and and a continuous mapping ${\ displaystyle K}$ ${\ displaystyle L}$

{\ displaystyle f \ colon \ vert K \ vert \ to \ vert L \ vert.}

A simplicial approximation of is a simplicial mapping ${\ displaystyle f}$

{\ displaystyle \ phi \ colon K \ to L}

with the property that for all the point lies in the closed carrier simplex of . ${\ displaystyle x \ in \ vert K \ vert}$ ${\ displaystyle \ vert \ phi \ vert (x) \ in \ vert L \ vert}$ ${\ displaystyle f (x)}$

Existence of simplicial approximations

In general, there does not have to be a simplicial approximation for a continuous mapping. But there is a simplicial approximation based on a sufficiently fine subdivision of the archetype complex . ${\ displaystyle K}$

Simplicial approximation theorem : For every continuous mapping there is a natural number , so that has a simplicial approximation. ${\ displaystyle f \ colon \ vert K \ vert \ to \ vert L \ vert}$ ${\ displaystyle q}$ ${\ displaystyle f \ colon \ vert K ^ {(q)} \ vert \ to \ vert L \ vert}$

Here the -th denotes the barycentric subdivision and it is known to apply . ${\ displaystyle K ^ {(q)}}$ ${\ displaystyle q}$ ${\ displaystyle \ vert K ^ {(q)} \ vert = \ vert K \ vert}$

An important step in proof is the following criterion: if there is a corner with every corner ${\ displaystyle x \ in K ^ {(q)}}$ ${\ displaystyle x ^ {\ prime} \ in L}$

{\ displaystyle f (\ operatorname {ost} (x)) \ subset \ operatorname {ost} (x ^ {\ prime})}

then the simplicial mapping defined by the assignment is a simplicial approximation of . Here is the open star of a corner . ${\ displaystyle x \ to x ^ {\ prime}}$ ${\ displaystyle \ phi}$ ${\ displaystyle f}$ ${\ displaystyle \ operatorname {east} (x)}$ ${\ displaystyle x}$

Homotopy

A simplicial approximation of a continuous mapping is too homotopic . This is because one can carry out the affine-linear homotopy between and within each closed simplex , and these homotopias coincide on the common side surfaces of closed simplices. ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle \ vert \ phi \ vert}$

Applications

By means of a simplicial approximation one obtains the functoriality of the simplicial homology with respect to continuous (instead of just simplicial) mappings. In particular, one obtains that homeoomorphic simplicial complexes have the same homology groups .

Brouwer used the approximation theorem to give rigorous proofs for the Jordan-Brouwer decomposition theorem and the theorem of invariance of dimension .

Furthermore, the isomorphism of singular and simplicial homology follows from the simplicial approximation theorem .

literature

Chapter 3.2 in: Ralph Stöcker, Heiner Zieschang : Algebraic Topology. Teubner, Stuttgart 1988, ISBN 3-519-02226-5