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## 数学代写|示性类代考Characteristic Classes代考|Relative cohomology and cohomology with coefficients of Lie algebras

Relative cohomology and cohomology with coefficients of Lie algebras. In the cohomology theory of topological spaces, there are notions of relative cohomology group $H^(X, A)$ of a pair $(X, A)$ and also cohomology group $H^(X ; \mathcal{S})$ with coefficients in a local system $\mathcal{S}$. They play important roles both theoretically and from the viewpoint of applications. Similarly, in the cohomology theory of Lie algebras, we have relative cohomology with respect to Lie subalgebras and also cohomology with twisted coefficients. We review these briefly.

Let $\mathfrak{g}$ be a Lie algebra. For any element $X \in \mathfrak{g}$ there is associated a linear map
$$i(X): C^k(\mathfrak{g}) \longrightarrow C^{k-1}(\mathfrak{g})$$
which is called the interior product and is defined by
$$(i(X) \varphi)\left(X_1, \cdots, X_{k-1}\right)=\varphi\left(X, X_1, \cdots, X_{k-1}\right)$$

## 数学代写|示性类代考Characteristic Classes代考|Cohomology of .5[(2, IR)

Cohomology of $\mathfrak{s}(2, \mathbb{R})$. As an example of cohomology of Lie algebras, here we compute the cohomology of the Lie algebra
$$\mathfrak{s l}(2, \mathbb{R})={X \in M(2, \mathbb{R}) ; \operatorname{Tr} X=0}$$
consisting of all $2 \times 2$ real matrices with $\mathrm{Tr}=0$. We choose
$$X_0=\left(\begin{array}{cc} 1 & 0 \ 0 & -1 \end{array}\right), X_1=\left(\begin{array}{ll} 0 & 1 \ 0 & 0 \end{array}\right), X_2=\left(\begin{array}{ll} 0 & 0 \ 1 & 0 \end{array}\right)$$
for a basis of $\mathfrak{s r}(2, \mathbb{R})$ and let
$$\varphi_0, \varphi_1, \varphi_2 \in C^1(\mathfrak{s l}(2, \mathbb{R}))=\mathfrak{s I}(2, \mathbb{R})^*$$
be its dual basis. Since
$$\left[X_0, X_1\right]:=2 X_1,\left[X_0, X_2\right]=-2 X_2,\left[X_1, X_2\right]=X_0,$$
we have
$$d \varphi_0=-\varphi_1 \wedge \varphi_2, d \varphi_1=-2 \varphi_0 \wedge \varphi_1, d \varphi_2=2 \varphi_0 \wedge \varphi_2 .$$
Hence
$$H^k\left(\boldsymbol { s } ( ( 2 , \mathbb { R } ) ) \quad \left{\begin{array}{ll} \mathbb{R} & k=0,3 \ 0 & k \neq 0,3, \end{array}\right.\right.$$
and we see that $\left[\varphi_0 \varphi_1 \varphi_2\right]$ is a generator of $H^3(\mathfrak{s I}(2, \mathbb{R}))$.
Next we consider the maximal compact subgroup of $S L(2, \mathbb{R})$, which is $S O(2)$, and compute the relative cohomology
$$H^(\mathfrak{s l}(2, \mathbb{R}), S O(2)) \quad H^(\mathfrak{s l}(2, \mathbb{R}), \mathfrak{s o}(2))$$
with respect to it. We can take $X=-X_1+X_2$ as a basis of $\mathfrak{s o}(2)$. Then we have
$$i(X) \varphi_0=0, i(X) \varphi_1=-1, i(X) \varphi_2=1 .$$

## 数学代写|示性类代考Characteristic Classes代考|Flat bundles

2.1.1. 陈-魏尔理论。让 $G$ 是一个李群并且让 $\pi: P \rightarrow M$ 当校长 $G$-㧽绑在 $C^{\infty}$ 歧管 $M$. 即珨出一个正确的动作
$$P \times G \longrightarrow P$$

Local triviality: 对于任何一点 $p \in M$ ，存在一个开邻域 $U \ni p$ 和微分同顺 $\varphi: \pi^{-1}(U) \cong U \times G$ 这样
$$\pi(u g)=\pi(u), \varphi(u g)=\varphi(u) g \quad\left(u \in \pi^{-1}(U), g \in G\right) .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|示性类代考Characteristic Classes代考|Flat bundles

2.1.1. Chern-Weil theory. Let $G$ be a Lie group and let $\pi$ : $P \rightarrow M$ be a principal $G$-bundle over a $C^{\infty}$ manifold $M$. Namely there is given a right action
$$P \times G \longrightarrow P$$
of the structure group $G$ on the total space $P$ satisfying the following condition.

Local triviality: For any point $p \in M$, there exist an open neighborhood $U \ni p$ and a diffeomorphism $\varphi: \pi^{-1}(U) \cong U \times G$ such that
$$\pi(u g)=\pi(u), \varphi(u g)=\varphi(u) g \quad\left(u \in \pi^{-1}(U), g \in G\right) .$$
For example, the tangent frame bundle $\pi: P(M) \rightarrow M$ of $M$ becomes a principal bundle with structure group $G L(n, \mathbb{R})$, where $\operatorname{dim} M=n$, and it is a very important principal bundle for the investigation of the structure of $M$. In fact, for the study of manifolds, it is one of the main tools to consider various bundles over $M$, not merely the tangent frame bundle, and then to examine the structure of them.

## 数学代写|示性类代考Characteristic Classes代考|Definition of flat bundles

Definition 2.1. A connection $\omega$ on a principal $G$-bundle is called a flat connection if its curvature $\Omega$ is identically 0 . A principal $G$ bundle equipped with a flat connection is called a flat $G$-bundle.
EXAMPLE 2.2. If we put the trivial connection on a product bundle $M \times G$, it is clearly a flat bundle. This is called a trivial flat bundle. The connection form $\omega_0$ of this bundle is given by $\omega_0=q^* \theta$ where $q: M \times G \rightarrow G$ is the natural projection and $\theta \in A^1(G ; \mathfrak{g})$ denotes the Maurer-Cartan form of $G$.

Example 2.3. Let $\pi: P \rightarrow M$ be a flat $G$-bundle and let $f:$ $N \rightarrow M$ be a $C^{\infty}$ map. Then the pullback bundle $f^* P \rightarrow N$ by $f$ becomes a flat $G$-bundle.

By virtue of the Chern-Weil theory, which we recalled in the previous subsection, any real characteristic class of a flat bundle vanishes. However, such bundle is not necessarily a trivial bundle as a principal bundle and furthermore, even if it were so, the flat connection on it is not necessarily a trivial one. Depending on the base space $M$, it may happen that there are many flat $G$-bundles on it. In such a situation, it often becomes an important problem to consider all flat bundles on $M$ and then classify them. Accordingly we first give a criterion of classification of flat bundles.

## 数学代写|示性类代考Characteristic Classes代考|Flat bundles

2.1.1. 陈-魏尔理论。让 $G$ 是一个李群并且让 $\pi: P \rightarrow M$ 当校长 $G$-㧽绑在 $C^{\infty}$ 歧管 $M$. 即珨出一个正确的动作
$$P \times G \longrightarrow P$$

Local triviality: 对于任何一点 $p \in M$ ，存在一个开邻域 $U \ni p$ 和微分同顺 $\varphi: \pi^{-1}(U) \cong U \times G$ 这样
$$\pi(u g)=\pi(u), \varphi(u g)=\varphi(u) g \quad\left(u \in \pi^{-1}(U), g \in G\right) .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 示性类

## avatest™帮您通过考试

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## 数学代写|示性类代考Characteristic Classes代考|Proof of the uniqueness of minimal models

Proof of the uniqueness of minimal models. In this subsection we prove the latter part of Theorem 1.41, namely the uniqueness of minimal models. Let us recall the precise statement. Let $\mathcal{A}$ be a cohomologically connected d.g.a and assume that we are given two minimal models $\rho: \mathcal{M} \rightarrow \mathcal{A}$ and $\rho^{\prime}: \mathcal{M}^{\prime} \rightarrow \mathcal{A}$. Then our task is to prove that there exists an isomorphism $\varphi: \mathcal{M} \cong \mathcal{M}^{\prime}$ such

that the diagram
\begin{aligned} &\mathcal{M} \stackrel{\rho}{\longrightarrow} \mathcal{A} \ &\begin{array}{ll} \varphi \downarrow & \ \mathcal{M}^{\prime}-\underset{\rho^{\prime}}{ } \longrightarrow & \mathcal{A} \end{array} \ & \end{aligned}
is commutative up to homotopy. Moreover we have to prove also that such map $\varphi$ is unique up to homotopy. By definition, $\mathcal{M}$ is generalized nilpotent so that it can be expressed as the union of certain increasing series
$$\text { (1.4) } \mathcal{M}0=K \subset \mathcal{M}_1 \subset \mathcal{M}_2 \subset \cdots \subset \mathcal{M}{\ell} \subset \cdots$$
of Hirsch extensions. It is natural to try to construct the desired map $\varphi$ by induction on $\ell$. Then there arise certain extension problems of d.g.a. maps. More precisely, we will apply the following proposition where we replace $\mathcal{N}, \mathcal{A}, f, \mathcal{B}$ in the statement by $\mathcal{M}_{\ell}, \mathcal{M}^{\prime}, \rho^{\prime}, \mathcal{A}$ respectively.

## 数学代写|示性类代考Characteristic Classes代考|Differential forms on simplicial complexes

Differential forms on simplicial complexes. The de Rham complex $A^*(M)$ of a $C^{\infty}$ manifold $M$ is a d.g.a. over $\mathbb{R}$ so that in principle we cannot deduce information on the rational homotopy type of $M$ from it. If there are given enough cycles of $M$ over $\mathbb{Z}$, then by investigating values of integrals over them, we can decide whether a given closed form represents a rational cohomology class or not. However, this procedure cannot be considered as an intrinsic structure of the de Rham complex. Then there appeared the de Rham theory for simplicial complexes which turn out to be utilized to obtain information about their structure over $\mathbb{Q}$.

To define the de Rham complex of simplicial complexes, first consider the $k$-dimensional standard simplex
$$\Delta^k=\left{\left(t_0, t_1, \cdots, t_k\right) \in \mathbb{R}^{k+1} ; t_i \geq 0, \sum_i t_i=1\right}$$

## 数学代写|示性类代考Characteristic Classes代考|Proof of the uniqueness of minimal models

$$\mathcal{M} \stackrel{\rho}{\longrightarrow} \mathcal{A} \quad \varphi \downarrow \quad \mathcal{M}^{\prime}-{ }{\rho^{\prime}} \longrightarrow \mathcal{A}$$ $$\text { (1.4) } \mathcal{M} 0=K \subset \mathcal{M}_1 \subset \mathcal{M}_2 \subset \cdots \subset \mathcal{M} \ell \subset \cdots$$ 应用以下命题 $\mathcal{N}, \mathcal{A}, f, \mathcal{B}$ 在声明中 $\mathcal{M}{\ell}, \mathcal{M}^{\prime}, \rho^{\prime}, \mathcal{A}$ 分别。

## 数学代写|示性类代考Characteristic Classes代考|Differential forms on simplicial complexes

〈left 缺少或无法识别的分隔符

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 示性类

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|示性类代考Characteristic Classes代考|Homology theory and homotopy theory

Homology theory and homotopy theory. Among the methods of investigating geometrical properties of a given figure or space, we have homology theory and homotopy theory. The former was begun by Poincaré around 1900, and the latter was initiated by Hurewicz in the 1930’s. Briefly speaking, in homology theory we decompose a given figure into components like points, segments, triangles and in general $k$-dimensional simplices (the triangulation), and then we extract a topological invariant called the homology group out of the way they are connected to each other. There is another way of decomposing figures, namely by means of CW complexes which are more flexible than triangulations. Just to make sure, let us recall the definition of CW complexes. Let $D^n=\left{x \in \mathbb{R}^n ;|x| \leq 1\right}$ be the $n$-dimensional disk and let $S^{n-1}=\partial D^n=\left{x \in \mathbb{R}^n ;|x|=1\right}$ be its boundary, namely the $(n-1)$-dimensional sphere.

DEFINITION 1.1. Let $X$ be a topological space and let $f: S^{n-1} \rightarrow$ $X$ be a continuous map. We denote by
$$X \cup_f D^n$$
the space obtained from the disjoint union of $X$ and $D^n$ by identifying each point $x \in S^{n-1}$ with $f(x) \in X$. It is called the space obtained by attaching an $n$-cell $e^n=D^n \backslash S^{n-1}$ to $X$ by $f$ or simply the attaching space (see Figure 1.1). The map $f$ is called the attaching map.

## 数学代写|示性类代考Characteristic Classes代考|Postnikov decomposition

Postnikov decomposition. Given a topological space $X$, if we can construct a triangulation of it or a decomposition as a CW complex, then it is convenient for the study of homological structure. However, it is not so useful for homotopy theoretical study of the space. The Postnikov decomposition describes the homotopy type of $X$ in terms of Eilenberg-MacLane spaces as fundamental components. The simplest space whose homotopy groups are the same as those of $X$ would be
$$K\left(\pi_1(X), 1\right) \times K\left(\pi_2(X), 2\right) \times \cdots$$
which is the product of various Eilenberg-MacLane spaces. In general, $X$ is not the product but a certain twisted version of the above space. The way it is twisted is described by what is called the Postnikov invariants.

## 数学代写|示性类代考Characteristic Classes代考|Homology theory and homotopy theory

〈left 缺少或无法识别的分隔符

$$X \cup_f D^n$$

## 数学代写|示性类代考Characteristic Classes代考|Postnikov decomposition

Postnikov分解。给定一个拓扑空间 $X$ ，如果戔们可以将它构造一个三角剖分或者分解为一个CW貪形，那/对于同调结构的研究 就很方便了。然而，它对空间的同伦理论研究没有多大用处。Postnikov 分解渵术了的同伦类型 $X$ 在 Eilenberg-MacLane 空间 方面作为基本组成部分。同伦群相同的最简单空间 $X$ 将会
$$K\left(\pi_1(X), 1\right) \times K\left(\pi_2(X), 2\right) \times \cdots$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。