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Published in:
2016 | OriginalPaper | Chapter
4. Manifolds
Author : Torsten Wedhorn
Published in: Manifolds, Sheaves, and Cohomology
Publisher: Springer Fachmedien Wiesbaden
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Abstract
In this chapter we define a real (respectively complex) premanifold as a space together with a sheaf of functions that locally resembles \(\mathbb{R}^{n}\) (respectively \(\mathbb{C}^{n}\)) for some n together with its sheaf of \(C^{\alpha}\)-functions (\(\alpha\in\mathbb{N}_{0}\cup\{\infty,\omega\}\) fixed) (respective of holomorphic functions). Hence we first define the abstract notion of a space together with a sheaf of R-algebras for some ring R. These are called R-ringed spaces. Then a real (respectively complex) premanifold will be a special case of an \(\mathbb{R}\)-ringed (respectively a \(\mathbb{C}\)-ringed) space. A (real or complex) manifold is defined to be a premanifold whose underlying topological space is Hausdorff and second countable. This implies several other nice topological properties. In the last section we explain simple constructions for how to get new (pre)manifolds from old ones. The most important constructions will be the product and the gluing of (pre)manifolds.