Good cover (algebraic topology)

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. In algebraic topology, an open cover is called a good cover if all open sets in the cover and all intersections of finitely many open sets, $U_{\alpha _{1}\ldots \alpha _{n}}=U_{\alpha _{1}\ldots \alpha _{n-1}}\cap U_{\alpha _{n}}$ , are contractible (Petersen 2006).

The concept was introduced by Andre Weil in 1952 for differential manifolds, demanding the $U_{\alpha _{1}\ldots \alpha _{n}}$ to be differentiably contractible. A modern version of this definition appears in Bott & Tu (1982).

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.)

Example

The two-dimensional surface of a sphere $S^{2}$ 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.

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
Petersen, Peter (2006), Riemannian geometry, Graduate Texts in Mathematics, 171 (2nd ed.), New York: Springer, p. 383, ISBN 978-0387-29246-5, MR 2243772

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