For a topological manifold one often requires that the inhabited finite intersections are homeomorphic to an open ball. Similarly, for a smooth manifold one often requires that the finite inhabited intersections are diffeomorphic to an open ball.
Due to this subtly, it is instructive to make explicit the following definition:
Every paracompact smooth manifold admits a Riemannian metric, and for any point in a Riemannian manifold there is a geodesically convex neighborhood (any two points in the neighborhood are connected by a unique geodesic in the neighborhood, one whose length is the distance between the points; see for example the remark after lemma 10.3 (Milnor) page 59, or (do Carmo), Proposition 4.2). A nonempty intersection of finitely many such geodesically convex neighborhoods is also geodesically convex. The inverse of the exponential map based at any interior point of a geodesically convex open subset gives a diffeomorphism from this subset to a star-shaped open subset of . Indeed, the Gauss lemma shows that the tangent map of the exponential map is invertible. By definition of geodesic convexity the exponential map is injective, hence a diffeomorphism. As proved in ball, star-shaped open subsets of are diffeomorphic to , which completes the proof.
It is apparently a folk theorem that every geodesically convex open neighbourhood in a Riemannian manifold is diffeomorphic to a Cartesian space. For instance, this is asserted in the proof of Theorem 5.1 of (BottTu), which claims the existence of differentiable good open covers. But a complete proof in the literature is hard to find. See also the discussion of the references at ball.
Every smooth paracompact manifold of dimension admits a differentiably good open cover, def. 2, hence an open cover such that every non-empty finite intersection is diffeomorphic to the Cartesian space .
By (Greene) every paracompact smooth manifold admits a Riemannian metric with positive convexity radius . Choose such a metric and choose an open cover consisting for each point of the geodesically convex open subset given by the geodesic -ball at . Since the injectivity radius of any metric is at least it follows from the minimality of the geodesics in a geodesically convex region that inside every finite nonempty intersection the geodesic flow around any point is of radius less than or equal the injectivity radius and is therefore a diffeomorphism onto its image.
Moreover, the preimage of the intersection region under the geometric flow is a star-shaped region in the tangent space : because the intersection of geodesically convex regions is itself geodesically convex, so that for any with the whole geodesic segment for is also in the region.
So we have that every finite non-empty intersection of the is diffeomorphic to a star-shaped region in a vector space. By the results cited at ball (e.g. theorem 237 of (Ferus)) this star-shaped region is diffeomorphic to an .
The same holds true for subcategories such as
It is sufficient to check this in . We need to check that for a good open cover and any morphism, we get commuting squares
such that the form a good open cover of .
Now, while does not have all pullbacks, the pullback of an open cover does exist, and since is necessarily a continuous function this is an open cover . The need not be contractible, but being open subsets of a paracompact manifold, they are themselves paracompact manifolds and hence admit themselves good open covers .
Then the family of composites is clearly a good open cover of .
Every finite CW complex admits a good open cover.
Hopefully someone can find a clear reference to a proof. The assertion for finite CW complexes is found for example here (Topology of Tiling Spaces by Sadun, p. 37). It is not immediately clear from the remarks there what obstructions would exist to generalizing the assertion to all CW complexes.
As indicated at CW complex, every CW complex is homotopy equivalent to a simplicial complex, and simplicial complexes certainly admit good covers by taking open stars.
In (Osborne-Stern 69) the following discussion for sufficient conditions getting “close” to good open covers is discussed:
For such that then admits a cover by open balls and such that all nonempty intersections of the covering cells are (q−1)-connected.
A cover refines another cover if each map is some .
Each differentially good cover has a unique smallest refinement to a differentially good cover that is closed under intersection.
The following nPOV perspective on good open covers gives a useful general “explanation” for their relevance, which also explains the role of good covers in Cech cohomology generally and abelian sheaf cohomology in particular.
Let be a good open cover by open balls in the strong sense: such that every finite non-empty intersection is diffeomorphic to an .
This implies the statement by the characterization of cofibrant objects in the projective structure.
This has a useful application in the nerve theorem.
Notice that the descent condition for simplicial presheaves on CartSp at (good) covers is very weak, since all Cartesian spaces are topologically contractible, so it is easy to find the fibrant objects in the topological localization of for the canonical coverage of CartSp. The above observation says that for computing the -valued cohomology of a diffeological space , it is sufficient to evaluate on (the Cech nerve of) a good cover of .
We can turn this around and speak for any site of a covering family as being good if the corresponding Cech nerve is degreewise a coproduct of representables. In the projective model structure on simplicial presheaves on such good covers will enjoy the central properties of good covers of topological spaces.
Manfredo do Carmo, Riemannian geometry (trans. Francis Flaherty), Birkhäuser (1992)
John Milnor, Morse theory , Princeton University Press (1963)
R. Greene, Complete metrics of bounded curvature on noncompact manifolds Archiv der Mathematik Volume 31, Number 1 (1978)
RP Osborne and JL Stern. Covering Manifolds with Cells, Pacific Journal of Mathematics, Vol 30, No. 1, 1969.
MathOverflow, Proving the existence of good covers