# 数学代写|拓扑学代写Topology代考|MATH3402

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## 数学代写|拓扑学代写Topology代考|Winding number

Consider a two-dimensional surface $\mathcal{S}$ punctured at point $P$ (figure $5.4$ ). In other words, $\mathcal{S}$ has a hole due to the removal of point $P$ from the space. The remaining space is denoted $\mathcal{S}^{\prime}=\mathcal{S}-{P}$. Take some starting point $x_{0} \in \mathcal{S}^{\prime}$, and consider some closed path $\gamma: I \rightarrow \mathcal{S}^{\prime}$, where $I=[0,1]$ is the unit interval, with $\gamma(1)=\gamma(0)$. Drawing a line $L$ from $P$ to $x_{0}$, we may define the angle of any point $\gamma(s)$ on the curve from $L$. $\theta$ may be thought of as a coordinate on the circle, $S^{1}$, and as one progresses along $\gamma(s)$, this angle may not be single-valued, since $\theta$ and $\theta+2 \pi$ represent the same point. We therefore unwrap the circle to form a line $\mathbb{R}$, as in figure $5.5$, allowing $\theta$ to have any angle from 0 to $\infty$. (In the terminology of fiber bundles (chapter 4), the new $\mathbb{R}$-valued function $\tilde{\theta}(s)$ is a lift of the multivalued function $\theta(s)$. $\theta$ and $\tilde{\theta}$ are strictly speaking different functions, but henceforth we will simply denote both functions as $\theta$.)

As closed loop $\gamma(s)$ completes its circuit from $s=0$ to $s=1$, the angle $\theta(s)$ evolves from an initial value $\theta(0)$ to final value $\theta(1)$. The winding number of path $\gamma$ about point $P$ is then defined to be
$$n(\gamma, P)=\frac{1}{2 \pi}[\theta(1)-\theta(0)]=\frac{1}{2 \pi} \int_{0}^{1} \frac{d \theta}{d s} d s .$$
The winding number is clearly an integer, since when the curve returns to its starting point it must end up at an angle that differs from its initial value by an integer multiple of $2 \pi$. Equally clear is the intuitive meaning of $n$ : it counts the number of times the curve encloses $P$ before returning to its starting point, with $n>0$ for counterclockwise windings and $n<0$ for clockwise.

An important theorem provides the connection between winding number and homotopy: Two loops $\gamma_{1}$ and $\gamma_{2}$ in $\mathcal{S}^{\prime}$ are homotopic to each other if and only if they have the same winding number about $P$. In other words, the winding number can be used to label the first homotopy class.

This theorem can be generalized to more complicated situations. For example, $\mathcal{S}$ may be punctured at multiple points, $P_{j}$, for $j=1, \ldots, n$. Every curve will then have multiple winding numbers: there will be a winding number $n_{j}(\gamma)=n\left(\gamma, P_{j}\right)$ about each puncture. Then two loops in the punctured space will be homotopic if and only if their complete set of winding numbers $\left{n_{1}, \ldots, n_{n}\right}$ are the same.

## 数学代写|拓扑学代写Topology代考|Index of zero points of vector fields

Let $U$ be an open set of a topological space and let $V$ be a vector field on $U$. A zero of $\boldsymbol{V}$ is a point $P \in U$ where all the components of $\boldsymbol{V}$ vanish, $\boldsymbol{V}=0$. A zero at $P$ is isolated if there is a neighborhood of $P$ that contains no other zeros. Let $\mathcal{Z}$ denote the set of zeros and $U^{\prime}=U-\mathcal{Z}$ be the complementary set of points in $U$ where the field is nonvanishing.

Consider a closed loop $\gamma$ that encloses isolated zero $P$ a single time in the counterclockwise direction. We can define the vector field along the curve, $\boldsymbol{V}{\gamma}(s) \equiv \boldsymbol{V}(\gamma(s))$. The index, $\mathcal{I}{P}(\boldsymbol{V})=\operatorname{Index}_{P}(\boldsymbol{V})$, of the vector field $\boldsymbol{V}$ about zero $P$ is given by the number of rotations of the vector field $V_{\gamma}(s)$ about $P$ as one circulates counterclockwise along $\gamma(s)$. Examples are shown in figure 5.6. If the rotation of the vector is counterclockwise, the index is positive; for clockwise rotations it is negative. If $P$ is the only singular point, then $V_{\gamma}(s)$ is independent of the chosen path (as long as the path completes one circuit of $P$ ), so the dependence on $\gamma$ is usually dropped. If there are multiple singular points, then the index of the field is the sum of the indices at all of the singular points, $\mathcal{I}=\sum_{p} \mathcal{I}_{p}$

Vector fields can always be related to differential operators (see chapter 4), so that index theorems, such as the Atiyah-Singer theorem or Riemann-Roch theorem which involve various types of generalized indices, provide linkages between (i) differential equations on a space, (ii) the possible vector fields on the space, (iii) and the topology of the space (sce scction 8). In chapter 6 scveral topological numbers relevant to optics will be defined which can be viewed as indices of the type defined above.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Winding number

$$n(\gamma, P)=\frac{1}{2 \pi}[\theta(1)-\theta(0)]=\frac{1}{2 \pi} \int_{0}^{1} \frac{d \theta}{d s} d s .$$

left{{_{{1}, Vdots, n_{n}}right}} 是相同的。

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