### 数学代写|黎曼曲面代写Riemann surface代考|MATH3405

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• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|CONSTRUCTION OF FLOWS

In this section we construct some flows on the one dimensional manifold $Z$. These will be used in following sections to move relative homology cycles. We will take some care in the construction of the flows, to obtain technically useful properties.

Suppose that $g$ is a holomorphic function on $Z$, such as one of the functions $g_{i j}(z)=g_i(z)-g_j(z)$. We want to construct a flow $f(z, t)$ with the property that $f(z, 0)=z$, and $g(f(z, t))$ is “downwind” of $g(z)$ in a certain desired direction. In other words, the time derivative of $g(f(z, t))$ is contained in an angular sector of the form
$$S(\pm \delta) \stackrel{\text { def }}{=}\left{r e^{i \theta}: \theta \in[\pi-\delta, \pi+\delta]\right}$$
so $g(f(z, t))$ is contained in an angular sector of the form
$$S(g(z), \pm \delta) \stackrel{\text { def }}{=}\left{g(z)+r e^{i \theta}: \theta \in[\pi-\delta, \pi+\delta]\right} .$$
We would also like to insure that at $t=1$, the flow has the effect of moving $g(f(z, t))$ a certain distance away from $g(z)$. This will be possible unless critical points of $g$ are encountered first. We require some special behaviour as the flow moves past critical points. There will be a one dimensional subset $\Lambda \subset \mathrm{C}$, the union of paths which are approximately paths of steepest descent leading away from critical points of $g$. The flow $f$ will have the effect of moving points to $\Lambda$, and then along $\Lambda$ away from the critical points.

Recall that we are admitting the possibility of rotating the $t$ or $\zeta$ planes. This is the same as multiplying the function $g$ by $e^{i \theta}$. After making such a rotation, we can assume that the desired direction of flow is in the negative real direction. Note that $g(P)=0$ for any of the functions $g_{i j}$ considered. Thus rotation preserves $g(P)$.

Our construction of flows will make reference to four numbers, a choice of angular error $\delta$, a choice of small number $\sigma$, a choice of number $L$, and a choice of radius $R$. The number $L$ represents the minimum amount by which the real part of $g$ should be decreased by the flow, unless a critical point is encountered. The angular error represents the maximum allowed deviation from the negative real direction, for the direction in which $g(z)$ moves when $z$ is moved by the flow. The $\sigma$ is a small number which indicates what happens when the flow goes past a critical point.

## 数学代写|黎曼曲面代写Riemann surface代考|MOVING RELATIVE HOMOLOGY CHAINS

In this section we will describe a formalism for moving relative homology chains. We will form a double complex to calculate relative homology, and then consider homotopies in this complex. It will be done explicitly, so as to facilitate getting bounds.
$Z$ is a complex manifold of dimension one, the universal cover of the original Riemann surface $S$. We consider indices $I=\left(i_0, \ldots, i_n\right)$, saying $|I|=n$. For each such index let $Z_I$ be the space $Z^n$. Let
$$Z_n=\coprod_{|I|=n} Z_I, \quad Z_*=\coprod_I Z_I .$$
We will work with chains which are combinations of singular and de Rham chains. Our manifolds will have linear structures, in other words embeddings as open sets in vector spaces. By a $k$-chain on such a manifold $Y$ we will mean a linear functional on the space of $C^{\infty}$ differential $k$-forms on $Y$ which can be expressed as a sum of components of the following form $h(u * H)$. Here $H$ is a $k+l$ dimensional space, compact, with linear structure and algebraic boundary, together with $h: H \rightarrow Y$ a smooth algebraic map (in other words the map is given by coordinate functions which are algebraic over the ring of polynomial functions on $H$ ). It is contracted with a smooth differential $l$-form $u$ on $H$. Such a chain provides a linear functional on the space of $k$-forms $a$ by the rule
$$\langle h(u * H), a\rangle=\int_H u \wedge h^*(a) .$$
The reader may think primarily of singular chains (corresponding to the case when $u$ is just the function 1). The more general singular-de Rham chains arise because we use cutoff functions later in the argument. Still, we usually denote $\langle\eta, a\rangle$ by $\int_\eta a$.

These algebraic singular-de Rham chains are functorial with respect to continuous piecewise polynomial maps (even though more general types of currents are not). Suppose $f: Y \rightarrow Y^{\prime}$ is continuous and piecewise polynomial, and suppose $h(u * H)$ is a $k$-chain on $Y$. The composition $f h: H \rightarrow Y^{\prime}$ is continuous and piecewise polynomial. We may further subdivide $H$ into finitely many pieces $H_i$ (with algebraic boundaries) such that on each $H_i, f h$ is polynomial. Let $u_i$ be the restriction of $u$ to $H_i$. Then define
$$f(h(u * H))=\sum(f h)\left(u_i * H_i\right)$$

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|MOVING RELATIVE HOMOLOGY CHAINS

$Z$ 是一维复流形，原黎曼曲面的普覆盖 $S$. 我们考虑指数 $I=\left(i_0, \ldots, i_n\right)$ ，说 $|I|=n$. 对于每个这样的 索引让 $Z_I$ 成为空间 $Z^n$. 让
$$Z_n=\coprod_{|I|=n} Z_I, \quad Z_*=\coprod_I Z_I$$

$$\langle h(u * H), a\rangle=\int_H u \wedge h^*(a) .$$

$$f(h(u * H))=\sum(f h)\left(u_i * H_i\right)$$

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