MATH528_Study_Guide - 1 1.1 Differential Topology/Geometry...

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1 Differential Topology/Geometry 1.1 Manifolds and Differentiable Structures n -Manifold. Let X be a topological space that is Hausdorff with a countable basis. Then X is a topological manifold if for all x X , there exists a set U 3 x such that there is a homeomophism ϕ : U D n n . We write X n for an n -manifold. The sets and functions U and φ are called a local chart . Atlas. Let X be a topological manifold. A collection of charts { U α , ϕ α } that covers X is called an atlas . The atlas { U α , ϕ α } is a smooth atlas if for all α, β , the map ϕ β ϕ - 1 α is smooth (i.e. in C ). Two atlases are equivalent if their union is still a smooth atlas. A smooth manifold is an equivalence class of atlases. The maps ϕ β ϕ - 1 α are called the transition maps . Examples of Manifolds. 0. n with U α open balls. 1. S n with an atlas of 2 n + 2 charts that are simply the hemispheres. S n could have an atlas of only 2 charts, namely the stereographic projections from the North and South poles.
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