Good cover (algebraic topology)

The cover on the left is not a good cover, since while all open sets in the cover are contractible, their intersection is disconnected. The cover on the right is a good cover, since the intersection of the two sets is contractible.

In mathematics, an open cover of a topological space X is a family of open subsets such that X is the union of all of the open sets. A good cover is an open cover in which all sets and all non-empty intersections of finitely-many sets are contractible (Petersen 2006).

The concept was introduced by André Weil in 1952 for differentiable manifolds, demanding the Uα1…αn to be differentiably contractible. A modern version of this definition appears in (Bott Tu).

## Application

A major reason for the notion of a good cover is that the Leray spectral sequence of a fiber bundle degenerates for a good cover, and so the Čech cohomology associated with a good cover is the same as the Čech cohomology of the space. (Such a cover is known as a Leray cover.) However, for the purposes of computing the Čech cohomology it suffices to have a more relaxed definition of a good cover in which all intersections of finitely many open sets have contractible connected components. This follows from the fact that higher derived functors can be computed using acyclic resolutions.

## Example

The two-dimensional surface of a sphere S2 has an open cover by two contractible sets, open neighborhoods of opposite hemispheres. However these two sets have an intersection that forms a non-contractible equatorial band. To form a good cover for this surface, one needs at least four open sets. A good cover can be formed by projecting the faces of a tetrahedron onto a sphere in which it is inscribed, and taking an open neighborhood of each face. The more relaxed definition of a good cover allows us to do this using only three open sets. A cover can be formed by choosing two diametrically opposite points on the sphere, drawing three non-intersecting segments lying on the sphere connecting them and taking open neighborhoods of the resulting faces.

## References

* Bott, Raoul; Tu, Loring (1982), Differential Forms in Algebraic Topology, New York: Springer, ISBN 0-387-90613-4 , §5, S. 42.
* Weil, Andre (1952), "Sur les theoremes de de Rham", Commentarii Math. Helv. 26: 119–145, doi:10.1007/BF02564296
* Petersen, Peter (2006), Riemannian geometry, Graduate Texts in Mathematics, 171 (2nd ed.), New York: Springer, p. 383, ISBN 978-0387-29246-5, https://books.google.com/books?id=9cekXdo52hEC&pg=PA383

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