Complex Manifolds and Vector Bundles

Abstract

An n-dimensional topological manifold is a Hausdorff topological space M such that every point p∊M has an open neighborhood homeomorphic to an open subset of the n-dimensional number space ℝⁿ. A coordinate covering {U _α , z _α } of such a manifold M consists of a covering of M by open subsets U _α together with homeomorphisms z _α :U _α →V _α between the sets U _α and open subsets V _α ⊆ℝⁿ; the sets U _α are called coordinate neighborhoods and the mappings z _α are called coordinate mappings. A topological manifold of course always admits coordinate coverings. Note that on each nonempty intersection U _α ⋂U _β of coordinate neighborhoods there are thus two homeomorphisms into ℝⁿ; the compositions

$${f_{\alpha \beta }} = {z_\alpha }\,o\,z_\beta ^{ - 1}:{z_\beta }\left( {{U_\alpha } \cap {U_\beta }} \right) \to {z_\alpha }\left( {{U_\alpha } \cap {U_\beta }} \right)$$

are called the coordinate transition functions of the coordinate covering, and for any point p∊U _α ⋂U _β the two coordinate mappings are related by z _α (p)=f _αβ (z _β (p)). The manifold M is completely determined by the sets {V _α } and the mappings {f _αβ }; for M can be obtained from the disjoint union of all the sets V _α by identifying a point z _α ∊V _α and a point z _β ∊V _β whenever z _α =f _αβ (z _β ).