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# 数学代写|微分拓扑作业代写differential topology代考|Smooth Manifolds

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## 数学代写|微分拓扑作业代写differential topology代考|Topological Manifolds

Let us get straight to our object of study. The terms used in the definition are explained immediately below the box. If words like “open” and “topology” are new to you, you are advised to read Appendix A on point-set topology in parallel with this chapter.

The last point (locally homeomorphic to $\mathbf{R}^{n}$ – implicitly with the metric topology – also known as Euclidean space, see Definition A.1.8) means that for every point $p \in M$ there is
an open neighborhood $U$ of $p$ in $M$,
an open set $U^{\prime} \subseteq \mathbf{R}^{n}$ and
a homeomorphism (Definition A.2.5) $x: U \rightarrow U^{\prime}$.
We call such an $x: U \rightarrow U^{\prime}$ a chart and $U$ a chart domain (Figure 2.1).

## 数学代写|微分拓扑作业代写differential topology代考|Smooth Structures

We will have to wait until Definition 2.3 .5 for the official definition of a smooth manifold. The idea is simple enough: in order to do differential topology we need that the charts of the manifolds are glued smoothly together, so that our questions regarding differentials or the like do not get different answers when interpreted through different charts. Again “smoothly” must be borrowed from the Euclidean world. We proceed to make this precise.

Let M be a topological manifold, and let $x_{1}: U_{1} \rightarrow U_{1}^{\prime}$ and $x_{2}: U_{2} \rightarrow U_{2}^{\prime}$ be two charts on $M$ with $U_{1}^{\prime}$ and $U_{2}^{\prime}$ open subsets of $\mathbf{R}^{n}$. Assume that $U_{12}=U_{1} \cap U_{2}$ is nonempty.

## 数学代写|微分拓扑作业代写DIFFERENTIAL TOPOLOGY代考|Maximal Atlases

We easily see that some manifolds can be equipped with many different smooth atlases. An example is the circle. Stereographic projection gives a different atlas than what you get if you for instance parametrize by means of the angle Example 2.2 .7 vs. Exercise 2.2.12. But we do not want to distinguish between these two “smooth structures”, and in order to systematize this we introduce the concept of a maximal atlas.

Let $M$ be a manifold and $\mathcal{A}$ a smooth atlas on $M$. Then we define $\mathcal{D}(\mathcal{A})$ as the following set of charts on $M$ :
$$\mathcal{D}(\mathcal{A})=\left{\begin{array}{l|l} \text { charts } y: V \rightarrow V^{\prime} \text { on } M & \begin{array}{c} \text { for all charts }(x, U) \text { in } \mathcal{A}, \text { the composite } \ \left.x\right|{W}\left(\left.y\right|{W}\right)^{-1}: y(W) \rightarrow x(W) \ \text { is a diffeomorphism, where } W=U \cap V \end{array} \end{array}\right}$$

## 数学代写|微分拓扑作业代写DIFFERENTIAL TOPOLOGY代考|MAXIMAL ATLASES

$$\mathcal{D}(\mathcal{A})=\left{\begin{array}{l|l} \text { charts } y: V \rightarrow V^{\prime} \text { on } M & \begin{array}{c} \text { for all charts }(x, U) \text { in } \mathcal{A}, \text { the composite } \ \left.x\right|{W}\left(\left.y\right|{W}\right)^{-1}: y(W) \rightarrow x(W) \ \text { is a diffeomorphism, where } W=U \cap V \end{array} \end{array}\right}$$

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