Glueing

Glueing in differential topology, algebraic and analytic geometry, etc. is a frequently used method to construct global objects such as varieties, schemes, differentiable manifolds, vector bundles, sheaves from local pieces from some category of local models together with glueing data.

Consider, for example, the case of differentiable manifolds of dimension . In this case the local model category consists of open sets in and differentiable mappings. A local-pieces-and-glueing-data description of an -dimensional differentiable manifold now consists of the following:

i) a collection of open subsets of indexed by ;

ii) for each and open subsets and together with a diffeomorphism .

The glueing data are subject to the following consistency conditions:

iii) , ;

iv) ;

v) on .

From these data one constructs a locally Euclidean space by taking the disjoint union, , of the modulo the equivalence relation if , and for some . If the resulting topological space is Hausdorff and paracompact, then a differentiable manifold is obtained. Both these properties do not follow from the construction. Local coordinate systems are obtained from (the inverses of) the natural mappings .

For (pre-)schemes the local model category is that of affine schemes and morphisms of schemes between them. Cf. Scheme. Here also global separation properties must be added to obtain a scheme. For vector bundles the local model category is that of trivial vector bundles , , and vector bundle morphisms between such trivial vector bundles, i.e. differentiable mappings of the form , where is an matrix smoothly depending on . Cf. Vector bundle.

If is a differentiable manifold with a covering by coordinate neighbourhoods and corresponding coordinate systems , then the corresponding local-pieces-and-glueing-data description of is as follows. The local pieces are the . The open subsets are equal to the and the glueing data are the mappings restricted to . Thus, the description of a manifold by means of an atlas and the description by means of local pieces and glueing data are quite close to one another.

References

[a1]	A. Grothendieck, "Elements de géométrie algébrique I" , IHES (1960) pp. Sect. 0.4.1.7
[a2]	M. Hazewinkel, "A tutorial introduction to differentiable manifolds and calculus on differentiable manifolds" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 316–340