Good coverage

In mathematics , good coverages are an aid in the topology of manifolds . Good coverages are coverages by open sets, the finite averages of which are all contractible .

Definition of manifolds

Let it be a manifold and a cover of by open sets . The coverage is good coverage if all finite averages ${\displaystyle M}$ ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle M}$ ${\displaystyle U_{i}}$

${\displaystyle U_{i_{0}}\cap \ldots \cap U_{i_{p}}}$

(with ) diffeomorph to be. ${\displaystyle i_{0},\ldots ,i_{p}\in I}$ ${\displaystyle \mathbb {R} ^{n}}$

existence

Every paracompact differentiable manifold has good coverage. If the manifold is compact , there are finite good covers.

Evidence sketch

Choose a Riemannian metric on the manifold. In a Riemannian manifold, every point has a geodetically convex neighborhood and the intersections of geodetically convex neighborhoods are again geodetically convex. A geodetically convex subset of a -dimensional Riemannian manifold is diffeomorphic to , from which the claim follows. ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$

Cofinality

The good coverages are cofinal in the set of all coverages, i.e. H. for every cover there is a refinement that is a good cover.

Applications

Good coverage is required for proofs using "Mayer-Vietoris arguments"; H. if statements are first to be proven locally and then globalized using Mayer-Vietoris sequences . Typical examples are the proofs of the Poincaré duality , the Künneth formula , the Leray-Hirsch theorem and the Thom isomorphism .

Definition of topological spaces

Let it be a topological space and a covering of by open sets . The coverage is good coverage if all finite averages ${\displaystyle X}$ ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle X}$ ${\displaystyle U_{i}}$

${\displaystyle U_{i_{0}}\cap \ldots \cap U_{i_{p}}}$

are (with ) contractible . ${\displaystyle i_{0},\ldots ,i_{p}\in I}$

(This definition is somewhat more general than the one given above in the special case of manifolds. However, this makes no difference in the case of paracompact manifolds, because every paracompact manifold has a good coverage even in the above sense.)

Not every topological space has a good cover. For example, there are topological spaces in which not every point has a contractible environment.

However, in the case of simplicial complexes , it is true that there are always good coverages and that the good coverages are cofinal in the set of all coverages.

literature

• Bott, Raoul; Tu, Loring W .: Differential forms in algebraic topology. Graduate Texts in Mathematics, 82nd Springer-Verlag, New York-Berlin, 1982. ISBN 0-387-90613-4

Web links

• good open cover in nlab

Individual evidence

1. ↑ Bott-Tu, op.cit., Theorem 5.1
2. ↑ cf. Bott-Tu, op.cit., P. 42ff
3. ↑ Bott-Tu, op.cit., P. 190