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 $ \dots $ from local pieces from some category of local models together with glueing data.

Consider, for example, the case of differentiable manifolds of dimension $ n $. In this case the local model category consists of open sets in $ \mathbf R ^ {n} $ and differentiable mappings. A local-pieces-and-glueing-data description of an $ n $- dimensional differentiable manifold $ M $ now consists of the following:

i) a collection of open subsets $ ( U _ \alpha ) $ of $ \mathbf R ^ {n} $ indexed by $ \alpha \in A $;

ii) for each $ \alpha $ and $ \beta $ open subsets $ U _ {\alpha \beta } \subset U _ \alpha $ and $ U _ {\beta \alpha } \subset U _ \beta $ together with a diffeomorphism $ \phi _ {\alpha \beta } : U _ {\alpha \beta } \rightarrow U _ {\beta \alpha } $.

The glueing data $ \phi _ {\alpha \beta } $ are subject to the following consistency conditions:

iii) $ U _ {\alpha \alpha } = U _ \alpha $, $ \phi _ {\alpha \alpha } = \mathop{\rm id} $;

iv) $ \phi _ {\alpha \beta } = \phi _ {\beta \alpha } ^ {-1} $;

v) $ \phi _ {\beta \gamma } \phi _ {\alpha \beta } = \phi _ {\alpha \gamma } $ on $ U _ {\alpha \gamma } \cap U _ {\alpha \beta } $.

From these data one constructs a locally Euclidean space $ M $ by taking the disjoint union, $ \amalg U _ \alpha $, of the $ U _ \alpha $ modulo the equivalence relation $ x \sim y $ if $ x \in U _ {\alpha \beta } $, $ y \in U _ {\beta \alpha } $ and $ \phi _ {\alpha \beta } ( x) = y $ for some $ \alpha , \beta $. If the resulting topological space $ M = \amalg U _ \alpha / \sim $ 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 $ U _ \alpha \rightarrow \amalg U _ \alpha \rightarrow M $.

For (pre-)schemes the local model category is that of affine schemes $ \mathop{\rm Spec} ( A) $ 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 $ U \times \mathbf R ^ {m} \rightarrow U $, $ U \subset \mathbf R ^ {n} $, and vector bundle morphisms between such trivial vector bundles, i.e. differentiable mappings $ U \times \mathbf R ^ {m} \rightarrow V \times \mathbf R ^ {m} $ of the form $ ( x, v) \mapsto ( \phi ( x), A ( x) v) $, where $ A ( x) $ is an $ m \times m $ matrix smoothly depending on $ x $. Cf. Vector bundle.

If $ M $ is a differentiable manifold with a covering $ ( V _ \alpha ) $ by coordinate neighbourhoods and corresponding coordinate systems $ \psi _ \alpha : V _ \alpha \rightarrow \mathbf R ^ {n} $, then the corresponding local-pieces-and-glueing-data description of $ M $ is as follows. The local pieces are the $ U _ \alpha = \psi _ \alpha ( V _ \alpha ) $. The open subsets $ U _ {\alpha \beta } $ are equal to the $ \psi _ \alpha ( V _ \alpha \cap V _ \beta ) $ and the glueing data $ \phi _ {\alpha \beta } : U _ {\alpha \beta } \rightarrow U _ {\beta \alpha } $ are the mappings $ \psi _ \beta \psi _ \alpha ^ {-1} $ restricted to $ \psi _ \alpha ( V _ \alpha \cap V _ \beta ) $. 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