19th Ave New York, NY 95822, USA

# 数学代写|代数拓扑作业代写algebraic topology代考|Homology with Local Coefficients

my-assignmentexpert™代数拓扑algebraic topology作业代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。my-assignmentexpert™， 最高质量的代数拓扑algebraic topology作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此代数拓扑algebraic topology作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

my-assignmentexpert™ 为您的留学生涯保驾护航 在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在代数拓扑algebraic topology代写方面经验极为丰富，各种代数拓扑algebraic topology相关的作业也就用不着 说。

## 数学代写|代数拓扑作业代写algebraic topology代考|Definition of homology with twisted coefficients

We begin with the definition of a group ring.
The group ring $\mathbf{Z} \pi$ is a ring associated to a group $\pi$. Additively it is the free abelian group on $\pi$; i.e. elements are (finite) linear combinations of the group elements
$$m_{1} g_{1}+\cdots+m_{k} g_{k} \quad m_{i} \in \mathbf{Z}, \quad g_{i} \in \pi .$$
Multiplication is given by the distributive law and multiplication in $\pi$ :
$$\left(\sum_{i} m_{i} g_{i}\right)\left(\sum_{j} n_{j} h_{j}\right)=\sum_{i, j}\left(m_{i} n_{j}\right)\left(g_{i} h_{j}\right) .$$

## 数学代写|代数拓扑作业代写algebraic topology代考|Examples and basic properties

The (ordinary) homology and cohomology groups are just special cases of the homology and cohomology with local coefficients corresponding to twisting by the trivial representations $\rho$ as we now show.
If ρ : π1X → Aut(A) is the trivial homomorphism, then the definition of tensor product gives a chain map
$$S_{}(\tilde{X}) \otimes_{\mathbf{Z} \pi} A_{\rho} \rightarrow S_{} X \otimes_{\mathbf{Z}} A$$
which we will see is an isomorphism. (In the chain complex on the left A is considered only as an abelian group.) This follows since both S∗(X>) and S∗(X) are chain complexes of free modules, so it is easy to compute tensor products. The complex S∗(X>) is a free Zπ-chain complex since π acts freely on X>, and hence on the set of all singular simplices in X>. We obtain a Zπ basis by choosing a representative simplex for each orbit. Better yet, for each singular simplex σ : ∆n → X, choose a single lift ˜σ : ∆n →X>. Then the set {σ˜} gives a basis for S∗(X>) over Zπ, and it follows that S∗(X>)⊗ZπAρ → S∗X⊗ZA is an isomorphism of graded abelian groups; from this description it is not hard to check that this isomorphism is a chain map, and so Hk(X; Aρ) = Hk(X; A), the usual homology with coefficients
in (the underlying Z-module) A.

## 数学代写|代数拓扑作业代写ALGEBRAIC TOPOLOGY代考|Definition of homology with a local coefficient system

The previous (algebraic) definition of homology and cohomology with local coefficients may appear to depend on base points, via the representation
$$\rho: \pi_{1}(X, *) \rightarrow \operatorname{Aut}(A),$$
and the identification of π1X with the covering translations of X>. In fact, it does not. We now give an alternative definition, which takes as input only the local coefficient system itself, i.e. the fiber bundle with discrete abelian group fibers. This definition is more elegant in that it does not depend on the arbitrary choice of a base point, but it is harder to compute with.

Let p : E → X be a system of local coefficients with fiber a discrete abelian group A and structure group G ⊂ Aut(A). Denote the fibers p−1(x) by Ex; for each x this is an abelian group non-canonically isomorphic to A.

We construct a chain complex as follows. Let Sk(X; E) denote the set of formal sums
$$\sum_{i=1}^{m} a_{i} \sigma_{i}$$
where:

1. . σi : ∆k → X is a singular k-simplex, and
2. ai is an element of the group Eσi(e0) where e0 ∈ ∆k is the base point (1, 0, 0, ··· , 0) of ∆k. More precisely, σi(e0) ∈ X, and we require ai ∈ Eσi(e0) = p−1(σi(e0)).

## 数学代写|代数拓扑作业代写ALGEBRAIC TOPOLOGY代考|DEFINITION OF HOMOLOGY WITH TWISTED COEFFICIENTS

(∑一世米一世G一世)(∑jnjHj)=∑一世,j(米一世nj)(G一世Hj).

## 数学代写|代数拓扑作业代写ALGEBRAIC TOPOLOGY代考|EXAMPLES AND BASIC PROPERTIES

$$S_{ }X~\otimes_{\mathbf{Z} \pi} A_{\rho} \rightarrow S_{ } X \otimes_{\mathbf{Z}} 我们将看到的$$

## 数学代写|代数拓扑作业代写ALGEBRAIC TOPOLOGY代考|DEFINITION OF HOMOLOGY WITH A LOCAL COEFFICIENT SYSTEM

ρ:圆周率1(X,∗)→或者⁡(一种),

∑一世=1米一种一世σ一世

1. . σi : ∆k → X 是一个奇异的 k-单纯形，并且
2. ai 是群 Eσi 的一个元素和0其中 e0 ∈ ∆k 是基点1,0,0,···,0Δk。更准确地说，σi和0∈ X，我们要求 ai ∈ Eσi和0= p−1σ一世(和0).

## 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。