## 数学代写|代数拓扑代写Algebraic topology代考|MATH354

statistics-lab™ 为您的留学生涯保驾护航 在代写代数拓扑Algebraic topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数拓扑Algebraic topology代写方面经验极为丰富，各种代写代数拓扑Algebraic topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|代数拓扑代写Algebraic topology代考|Operations on Spaces

Cell complexes have a very nice mixture of rigidity and flexibility, with enough rigidity to allow many arguments to proceed in a combinatorial cell-by-cell fashion and enough flexibility to allow many natural constructions to be performed on them. Here are some of those constructions.

Products. If $X$ and $Y$ are cell complexes, then $X \times Y$ has the structure of a cell complex with cells the products $e_\alpha^m \times e_\beta^n$ where $e_\alpha^m$ ranges over the cells of $X$ and $e_\beta^n$ ranges over the cells of $Y$. For example, the cell structure on the torus $S^1 \times S^1$ described at the beginning of this section is obtained in this way from the standard cell structure on $S^1$. For completely general CW complexes $X$ and $Y$ there is one small complication: The topology on $X \times Y$ as a cell complex is sometimes finer than the product topology, with more open sets than the product topology has, though the two topologies coincide if either $X$ or $Y$ has only finitely many cells, or if both $X$ and $Y$ have countably many cells. This is explained in the Appendix. In practice this subtle issue of point-set topology rarely causes problems, however.

Quotients. If $(X, A)$ is a CW pair consisting of a cell complex $X$ and a subcomplex $A$, then the quotient space $X / A$ inherits a natural cell complex structure from $X$. The cells of $X / A$ are the cells of $X-A$ plus one new 0-cell, the image of $A$ in $X / A$. For a cell $e_\alpha^n$ of $X-A$ attached by $\varphi_\alpha: S^{n-1} \rightarrow X^{n-1}$, the attaching map for the corresponding cell in $X / A$ is the composition $S^{n-1} \rightarrow X^{n-1} \rightarrow X^{n-1} / A^{n-1}$.

For example, if we give $S^{n-1}$ any cell structure and build $D^n$ from $S^{n-1}$ by attaching an $n$-cell, then the quotient $D^n / S^{n-1}$ is $S^n$ with its usual cell structure. As another example, take $X$ to be a closed orientable surface with the cell structure described at the beginning of this section, with a single 2-cell, and let $A$ be the complement of this 2-cell, the 1-skeleton of $X$. Then $X / A$ has a cell structure consisting of a 0 -cell with a 2-cell attached, and there is only one way to attach a cell to a 0-cell, by the constant map, so $X / A$ is $S^2$.
Suspension. For a space $X$, the suspension $S X$ is the quotient of $X \times I$ obtained by collapsing $X \times{0}$ to one point and $X \times{1}$ to another point. The motivating example is $X=S^n$, when $S X=S^{n+1}$ with the two ‘suspension points’ at the north and south poles of $S^{n+1}$, the points $(0, \cdots, 0, \pm 1)$. One can regard $S X$ as a double cone on $X$, the union of two copies of the cone $C X=(X \times I) /(X \times{0})$. If $X$ is a CW complex, so are $S X$ and $C X$ as quotients of $X \times I$ with its product cell structure, $I$ being given the standard cell structure of two 0 -cells joined by a 1-cell.

Suspension becomes increasingly important the farther one goes into algebraic topology, though why this should be so is certainly not evident in advance. One especially useful property of suspension is that not only spaces but also maps can be suspended. Namely, a map $f: X \rightarrow Y$ suspends to $S f: S X \rightarrow S Y$, the quotient map of $f \times \mathbb{1}: X \times I \rightarrow Y \times I$.

## 数学代写|代数拓扑代写Algebraic topology代考|Collapsing Subspaces

The operation of collapsing a subspace to a point usually has a drastic effect on homotopy type, but one might hope that if the subspace being collapsed already has the homotopy type of a point, then collapsing it to a point might not change the homotopy type of the whole space. Here is a positive result in this direction:
|f If $(X, A)$ is a CW pair consisting of a CW complex $X$ and a contractible subcomplex $A$, then the quotient map $X \rightarrow X / A$ is a homotopy equivalence.

A proof will be given later in Proposition 0.17, but for now let us look at some examples showing how this result can be applied.

Example 0.7: Graphs. The three graphs $0-\infty$ D are homotopy equivalent since each is a deformation retract of a disk with two holes, but we can also deduce this from the collapsing criterion above since collapsing the middle edge of the first and third graphs produces the second graph.

More generally, suppose $X$ is any graph with finitely many vertices and edges. If the two endpoints of any edge of $X$ are distinct, we can collapse this edge to a point, producing a homotopy equivalent graph with one fewer edge. This simplification can be repeated until all edges of $X$ are loops, and then each component of $X$ is either an isolated vertex or a wedge sum of circles.

This raises the question of whether two such graphs, having only one vertex in each component, can be homotopy equivalent if they are not in fact just isomorphic graphs. Exercise 12 at the end of the chapter reduces the question to the case of connected graphs. Then the task is to prove that a wedge sum $\bigvee_m S^1$ of $m$ circles is not homotopy equivalent to $V_n S^1$ if $m \neq n$. This sort of thing is hard to do directly. What one would like is some sort of algebraic object associated to spaces, depending only on their homotopy type, and taking different values for $\bigvee_m s^1$ and $V_n s^1$ if $m \neq n$. In fact the Euler characteristic does this since $V_m S^1$ has Euler characteristic $1-m$. But it is a rather nontrivial theorem that the Euler characteristic of a space depends only on its homotopy type. A different algebraic invariant that works equally well for graphs, and whose rigorous development requires less effort than the Euler characteristic, is the fundamental group of a space, the subject of Chapter 1 .

# 代数拓扑代考

## 数学代写|代数拓扑代写Algebraic topology代考|Operations on Spaces

$C X=(X \times I) /(X \times 0)$. 如果 $X$ 是 $\mathrm{CW}$ 复形，所以是 $S X$ 和 $C X$ 作为商 $X \times I$ 凭借其产品 细胞结构， $I$ 被陚予由一个 1 单元连接的两个 0 单元的标准单元结构。

## 数学代写|代数拓扑代写Algebraic topology代考|Collapsing Subspaces

|f 如果 $(X, A)$ 是由 $\mathrm{CW}$ 复合体组成的 $\mathrm{CW}$ 对 $X$ 和一个可收缩的子复合体 $A$ ，那么商图 $X \rightarrow X / A$ 是同伦等价。

$1-m$. 但空间的欧拉特征仅取决于其同伦类型是一个相当重要的定理。一个不同的代数不变 量同样适用于图，并且其严格的发展比欧拉特征需要更少的努力，是空间的基本群，第1 章的 主题。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|代数拓扑代写Algebraic topology代考|MATH8062

statistics-lab™ 为您的留学生涯保驾护航 在代写代数拓扑Algebraic topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数拓扑Algebraic topology代写方面经验极为丰富，各种代写代数拓扑Algebraic topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|代数拓扑代写Algebraic topology代考|Homotopy and Homotopy Type

One of the main ideas of algebraic topology is to consider two spaces to be equivalent if they have ‘the same shape’ in a sense that is much broader than homeomorphism. To take an everyday example, the letters of the alphabet can be written either as unions of finitely many straight and curved line segments, or in thickened forms that are compact regions in the plane bounded by one or more simple closed curves. In each case the thin letter is a subspace of the thick letter, and we can continuously shrink the thick letter to the thin one. A nice way to do this is to decompose a thick letter, call it $\mathbf{X}$, into line segments connecting each point on the outer boundary of $\mathbf{X}$ to a unique point of the thin subletter $\mathbf{X}$, as indicated in the figure. Then we can shrink $\mathbf{X}$ to $\mathbf{X}$ by sliding each point of $\mathbf{X}-\mathbf{X}$ into $X$ along the line segment that contains it. Points that are already in $X$ do not move.
We can think of this shrinking process as taking place during a time interval $0 \leq t \leq 1$, and then it defines a family of functions $f_t: \mathbf{X} \rightarrow \mathbf{X}$ parametrized by $t \in I=$ $[0,1]$, where $f_t(x)$ is the point to which a given point $x \in \mathbf{X}$ has moved at time $t$. Naturally we would like $f_t(x)$ to depend continuously on both $t$ and $x$, and this will be true if we have each $x \in \mathbf{X}-\mathrm{X}$ move along its line segment at constant speed so as to reach its image point in $\mathrm{X}$ at time $t=1$, while points $x \in \mathrm{X}$ are stationary, as remarked earlier.

Examples of this sort lead to the following general definition. A deformation retraction of a space $X$ onto a subspace $A$ is a family of maps $f_t: X \rightarrow X, t \in I$, such that $f_0=\mathbb{1}$ (the identity map), $f_1(X)=A$, and $f_t \mid A=\mathbb{1}$ for all $t$. The family $f_t$ should be continuous in the sense that the associated map $X \times I \rightarrow X,(x, t) \mapsto f_t(x)$, is continuous.

It is easy to produce many more examples similar to the letter examples, with the deformation retraction $f_t$ obtained by sliding along line segments. The figure on the left below shows such a deformation retraction of a Möbius band onto its core circle.

## 数学代写|代数拓扑代写Algebraic topology代考|Cell Complexes

A familiar way of constructing the torus $S^1 \times S^1$ is by identifying opposite sides of a square. More generally, an orientable surface $M_g$ of genus $g$ can be constructed from a polygon with $4 g$ sides by identifying pairs of edges, as shown in the figure in the first three cases $g=1,2,3$. The $4 g$ edges of the polygon become a union of $2 g$ circles in the surface, all intersecting in a single point. The interior of the polygon can be thought of as an open disk, or a 2-cell, attached to the union of the $2 g$ circles. One can also regard the union of the circles as being obtained from their common point of intersection, by attaching $2 g$ open arcs, or 1-cells. Thus the surface can be built up in stages: Start with a point, attach 1-cells to this point, then attach a 2-cell.

A natural generalization of this is to construct a space by the following procedure:
(1) Start with a discrete set $X^0$, whose points are regarded as 0-cells.
(2) Inductively, form the $n$-skeleton $X^n$ from $X^{n-1}$ by attaching $n$-cells $e_\alpha^n$ via maps $\varphi_\alpha: S^{n-1} \rightarrow X^{n-1}$. This means that $X^n$ is the quotient space of the disjoint union $X^{n-1} \amalg_\alpha D_\alpha^n$ of $X^{n-1}$ with a collection of $n$-disks $D_\alpha^n$ under the identifications $x \sim \varphi_\alpha(x)$ for $x \in \partial D_\alpha^n$. Thus as a set, $X^n=X^{n-1} \amalg_\alpha e_\alpha^n$ where each $e_\alpha^n$ is an open $n$-disk.
(3) One can either stop this inductive process at a finite stage, setting $X=X^n$ for some $n<\infty$, or one can continue indefinitely, setting $X=\bigcup_n X^n$. In the latter case $X$ is given the weak topology: A set $A \subset X$ is open (or closed) iff $A \cap X^n$ is open (or closed) in $X^n$ for each $n$.

# 代数拓扑代考

## 数学代写|代数拓扑代写Algebraic topology代考|Homotopy and Homotopy Type

㡾容易产生更多类似于字母示例的示例，变形收缩 $f_t$ 通过沿线段滑动获得。左下图显示了莫比 乌斯带在其核心圆上的变形收缩。

## 数学代写|代数拓扑代写Algebraic topology代考|Cell Complexes

(1) 从离散集开始 $X^0$ ，其点被视为 0 单元格。
(2) 归纳地，形成 $n$-骨骼 $X^n$ 从 $X^{n-1}$ 通过附加 $n$-细胞 $e_\alpha^n$ 通过地图 $\varphi_\alpha: S^{n-1} \rightarrow X^{n-1}$. 这意 味着 $X^n$ 是不相交并集的商空间 $X^{n-1} \amalg_\alpha D_\alpha^n$ 的 $X^{n-1}$ 与一系列 $n$-磁盘 $D_\alpha^n$ 在标识下 $x \sim \varphi_\alpha(x)$ 为了 $x \in \partial D_\alpha^n$. 因此作为一个集合， $X^n=X^{n-1} \amalg_\alpha e_\alpha^n$ 每个 $e_\alpha^n$ 是一个开放的 $n$ -磁盘。
(3) 可以在有限阶段停止这个归纳过程，设置 $X=X^n$ 对于一些 $n<\infty$ ，或者一个可以无限 期地继续，设置 $X=\bigcup_n X^n$. 在后一种情况下 $X$ 给定弱拓扑： A集合 $A \subset X$ 是开放的（或封 闭的) 当且仅当 $A \cap X^n$ 在中打开 (或关闭) $X^n$ 每个 $n$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|代数拓扑代写Algebraic topology代考|МА3403

statistics-lab™ 为您的留学生涯保驾护航 在代写代数拓扑Algebraic topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数拓扑Algebraic topology代写方面经验极为丰富，各种代写代数拓扑Algebraic topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|代数拓扑代写Algebraic topology代考|The shape of a space

Two topological spaces $X, Y$ are homeomorphic (written as $X \cong Y$ ) if there exists a homeomorphism $X \rightarrow Y$, i.e. a continuous mapping which has a continuous inverse. We also say that $X, Y$ have the same (topological) shape, or the same homeomorphism-type.

To prove that two given spaces have the same shape can require long computations, in order to build a homeomorphism between them, but is generally a ‘confined’ problem.

To prove that they are not homeomorphic can be quite difficult, and enticing, even when we clearly ‘see’ that they have a different shape: we must prove that there cannot exist a homeomorphism between them.
The proof is generally based on a ‘topological property’ (invariant up to homeomorphism) which holds in one of them but not in the other: connectedness, compactness, separation or countability axioms, higher forms of connectedness, etc. When these topological properties are explored throughout algebraic structures we enter in the domain of Algebraic Topology.

The following exercises show two families of spaces which is important to classify; the first classification can be achieved with the usual means of General Topology, the second will be completed later, using singular homology.

Exercises and complements. The solutions to these exercises can be found in Chapter 8.
(a) (The shape of the intervals, I) Any non-degenerate interval of the euclidean line is homeomorphic to $] 0,1[$, or $[0,1[$, or $[0,1]$.
Hints. It is an easy exercise of Calculus, which can be solved using affine linear functions and some elementary transcendental functions: see 8.1.1. (One can also use rational functions and their pastings, but the argument would be longer.)

## 数学代写|代数拓扑代写Algebraic topology代考|Classifying maps

Studying the maps $f: X \rightarrow Y$ between two topological spaces, we are also interested in classifying them, ‘up to continuous deformation’.

Let us recall that two maps $f, g: X \rightarrow Y$ are said to be homotopic (written as $f \simeq g$ ) if there is a map $\varphi: X \times \mathbb{I} \rightarrow Y$ defined on the cylinder $I(X)=X \times \mathbb{I}$ (with the product topology), that coincides with $f$ on the lower basis of the cylinder and with $g$ on the upper one
\begin{aligned} & \varphi: X \times \mathbb{I} \rightarrow Y, \ & \varphi(x, 0)=f(x), \quad \varphi(x, 1)=g(x) \quad(\text { for } x \in X), \end{aligned}
forming a continuous deformation of $f$ into $g$. This is easily proved to be an equivalence relation (see 1.4.4). The homotopy will also be written as $\varphi: f \simeq g: X \rightarrow Y$.

A point $x \in X$ can be identified with the corresponding map $x:{} \rightarrow X$, defined on the singleton space. A homotopy $a: x \simeq x^{\prime}:{} \rightarrow X$ is then the same as a path in $X$ from $x$ to $x^{\prime}$, i.e. a map such that
$$a: \mathbb{I} \rightarrow X, \quad a(0)=x, \quad a(1)=x^{\prime} .$$
On the other hand, a homotopy $\varphi: f \simeq g: X \rightarrow Y$ gives a family of paths $\varphi(x,-): \mathbb{I} \rightarrow Y$ from $f(x)$ to $g(x)$, indexed by $x \in X$ and varying continuously on $X$.

Exercises and complements. (a) The classification of the maps ${*} \rightarrow X$ up to homotopy amounts to the partition of $X$ in path components.
(b) We have two maps $f, g: X \rightarrow Y$ with values in a euclidean space $Y$ (i.e. a subspace of some $\mathbb{R}^n$ ).

If $Y$ is a convex subset of $\mathbb{R}^n$ (i.e. for all $y, y^{\prime} \in Y$, the line segment from $y$ to $y^{\prime}$ is contained in $Y$ ), the maps $f$ and $g$ are always homotopic. More generally, the same holds if, for every $x \in X$, the line segment from $f(x)$ to $g(x)$ is contained in $Y$.
Hints. We can use the linear structure of $\mathbb{R}^n$ to define the affine homotopy
$$\varphi: f \simeq g: X \rightarrow Y, \quad \varphi(x, t)=(1-t) f(x)+t g(x),$$
which describes, at each $x \in X$, the line segment from $f(x)$ to $g(x)$, and is indeed an affine linear map in the variable $t \in \mathbb{I}$.

# 代数拓扑代考

## 数学代写|代数拓扑代写Algebraic topology代考|The shape of a space

。 $] 0,1[$ ，或者 $[0,1[$ ，或者 $[0,1]$

(也可以使用有理函数及其粘贴，但论证会更长。)

## 数学代写|代数拓扑代写Algebraic topology代考|Classifying maps

$$\varphi: X \times \mathbb{I} \rightarrow Y, \quad \varphi(x, 0)=f(x), \quad \varphi(x, 1)=g(x) \quad(\text { for } x \in X),$$

$$a: \mathbb{I} \rightarrow X, \quad a(0)=x, \quad a(1)=x^{\prime} .$$

(b) 我们有两张地图 $f, g: X \rightarrow Y$ 具有欧几里得空间中的值 $Y$ (即一些子空间 $\mathbb{R}^n$ ).

$$\varphi: f \simeq g: X \rightarrow Y, \quad \varphi(x, t)=(1-t) f(x)+t g(x),$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|代数拓扑代写Algebraic topology代考|MX4546

statistics-lab™ 为您的留学生涯保驾护航 在代写代数拓扑Algebraic topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数拓扑Algebraic topology代写方面经验极为丰富，各种代写代数拓扑Algebraic topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|代数拓扑代写Algebraic topology代考|Translations and underlying affinities

As already said, Algebraic Topology is a ‘discipline of translations’, from topology to algebra. Each translation gives a partial view of spaces and maps, in a particular perspective. This simplified picture may be able to solve the problem we are studying; otherwise, another translation might do.

The grammar of this discipline is provided by elementary category theory, reviewed here in some sections of Chapter 1 , as far as this makes clear how the translation works.

As a simple instance, the fact that the functor $H_n$ transforms a disjoint union of spaces $\cup X_i$ into a direct sum $\oplus H_n\left(X_i\right)$ of abelian groups is the outer appearance of an underlying property, the preservation of categorical sums: a disjoint union of spaces and a direct sum of abelian groups satisfy the same categorical universal property (see 1.6.2).

The effectiveness of a translation likely depends on the affinity between the two languages involved, in our case the source category of topological spaces versus the algebraic category of destination. In this regard, the category Ab is sufficiently flexible to offer ‘covariant translations’, by homology theories, and ‘contravariant translations’, by cohomology theories.

Less flexible categories, like those of rings and graded algebras, can only give a ‘mirror-image’, by a cohomology functor $H^$. But this translation is far richer than viewing $H^(X)$ as a mere graded abelian group.

The roots of this phenomenon can be found in an affinity between Top and Rng ${ }^{\text {op }}$, the dual of the category of rings; an affinity expressed in theorems, like Gelfand duality, and in formal categorical aspects sketched in 5.5.9.

## 数学代写|代数拓扑代写Algebraic topology代考|An inductive approach on structural bases

Notions will be presented in a concrete, ‘inductive’ way, starting from elementary examples. Whenever possible, the reader will be guided to build the theory through a series of exercises.

On the other hand, the roots of the interplay between Topology and Algebra, formalised in Homological Algebra and Category Theory, will be investigated more deeply than usual in an introductory book.

Algebraic Topology is a complex domain, with ramifications in diverse fields. After covering the basic parts, and some more advanced ones, we shall sketch several developments that do not have a place here, giving extensive references for further study.

The exposition will be particularly elementary and detailed in Chapter 1 and the first part of Chapter 2: this is not a book on general topology or abelian groups, but the reader will be invited (and guided) to check the topological or algebraic ground on which we are building. This point made, we shall go on more quickly.

Chapters 1 and 2 form a basic introduction to singular homology; Chapters 1 and 6 play the same role for homotopy groups; the three of them can give a preliminary view of Algebraic Topology.

The only prerequisites are general topology and the very basic theory of groups, abelian groups and rings. Categories and functors are introduced when their need arises. The same holds for the part of Homological Algebra used here: exact sequences, chain complexes of abelian groups, tensor products and Hom-functors, their derived functors.

For the fascinating field of Algebraic Topology there are elementary textbooks, like Vick [Vi] and Massey [Mas3], and more advanced ones, like Hilton-Wylie [HiW], Spanier [Sp2] and Hatcher [Ha]; the last is freely downloadable. The history of this discipline is dealt with in Dieudonné [Di].

For Homological Algebra we shall refer to Cartan-Eilenberg [CE] and Mac Lane [M1]; for General Topology to Kelley [Ke], Munkres [Mu] and Bourbaki [Bou2].

Category Theory is exposed in well-known books, like Mac Lane [M2], Borceux [Bo1, Bo2, Bo3], Adámek, Herrlich and Strecker [AHS]. The author’s [G5] is a textbook for beginners, also devoted to applications in Algebra, Topology, Algebraic Topology and Homological Algebra.

# 代数拓扑代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|代数拓扑代写Algebraic topology代考|MAT9580

statistics-lab™ 为您的留学生涯保驾护航 在代写代数拓扑Algebraic topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数拓扑Algebraic topology代写方面经验极为丰富，各种代写代数拓扑Algebraic topology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|代数拓扑代写Algebraic topology代考|Homology and cohomology theories

Basically, a homology theory is a sequence of transformations $H_n(-)$; each of them turns a topological space $X$ into an abelian group $H_n(X)$, and a continuous mapping $f: X \rightarrow Y$ into a homomorphism $H_n(f): H_n(X) \rightarrow$ $H_n(Y)$. In this transformation, composition of maps and identity maps are preserved:

• for consecutive maps $f: X \rightarrow Y$ and $g: Y \rightarrow Z$, we have: $H_n(g f)=$ $H_n(g) H_n(f)$
• for a space $X$ we have: $H_n(\mathrm{id} X)=$ id $H_n(X)$.
In the basic terminology of category theory, each $H_n$ is a covariant functor, from the category Top of topological spaces (and continuous mappings) to the category Ab of abelian groups (and homomorphisms). As an obvious consequence, each $H_n$ preserves invertible arrows: it transforms a homeomorphism of spaces into an isomorphism of groups.

In fact, a homology theory turns up in a richer form: for every ‘relative pair’ $(X, A)$ of topological spaces (where $A$ is a subspace of $X$ ), and a fixed abelian group $G$, we have a sequence of groups $H_n(X, A ; G)$ of relative homology, with coefficients in $G$. The basic case is absolute homology with integral coefficients: $H_n(X)=H_n(X, \emptyset ; \mathbb{Z})$.

On the other hand, a cohomology theory $H^n(-)$ takes a continuous mapping $f: X \rightarrow Y$ to a homomorphism $H^n(f): H^n(Y) \rightarrow H^n(X)$, reversing the direction of arrows, reversing composition and preserving identities. Here also we have an enriched version, for relative pairs of topological spaces and a coefficient group $G$.

An important fact appears: the cohomology groups with coefficients in a ring $R$ have a graded multiplication, and their family forms a graded $R$-algebra. We shall study singular cohomology (a by-product of singular homology), and briefly review other theories, namely Alexander-Spanier cohomology and de Rham cohomology, which naturally arise in contravariant form.

## 数学代写|代数拓扑代写Algebraic topology代考|An outline

Chapter 1 is an introduction to the goals and methods of Algebraic Topology, with a brief analysis of the elementary issues of category theory involved in the interface between topology and algebra.

Chapter 2 introduces singular homology, the simplest homology theory of general topological spaces, showing how to compute the homology groups, and get information on spaces and maps. Section $2.4$ is devoted to the homology groups of the spheres, and their consequences: for instance, the Theorem of Topological Dimension, the topological degree of a map $\mathbb{S}^n \rightarrow \mathbb{S}^n$, and the theorem about vector fields on even-dimensional spheres. Several computations of homology groups can be found in Sections 2.5, 2.6.
Singular homology is constructed in the cubical form, that – in the author’s opinion – gives a simpler approach. The more usual simplicial form is briefly presented in Section 2.8; their equivalence is proved in Section 5.6.

In Chapter 3 we extend singular homology to relative pairs $(X, A)$. Section $3.3$ shows how relative homology is able to investigate local features, for instance the local and global orientation of topological manifolds. The Eilenberg-Steenrod axioms for relative homology theories are listed in Section 3.4, and verified for relative singular homology. Alexander-Spanier cohomology, described in Section 3.5, satisfies the dual axioms, for a relative cohomology theory; its graded product is dealt with in Section 3.6. Similarly, de Rham cohomology of differentiable manifolds gives a graded algebra on the real field (see Section 3.7).

Singular homology and cohomology with a coefficient group is studied in Chapters 4 and 5. Varying the coefficient group we may be able to get results that the ordinary theory (with integral coefficients) cannot obtain, as shown in Subsection 4.3.7. All this requires some tools of Homological Algebra, namely the tensor and torsion products for homology, or the Hom and Ext functors for cohomology (in Sections $4.1$ and 5.1-5.3).

Chapter 5 also explores the multiplicative structure of singular cohomology with coefficients in a ring (in Section 5.5), and the homology groups of a product of spaces (in Section 5.6). The latter are detemined by EilenbergMac Lane’s Acyclic Model Theorem, which is also used to prove the equivalence of the cubical and simplicial constructions of singular homology.
Chapter 6 is an introduction to the fundamental groupoid and homotopy groups, with their relationship to singular homology.

Chapter 7 briefly reviews issues that have already appeared in the previous chapters, on category theory (categorical limits and adjoint functors) and general topology (the compact-open topology). Finally Chapter 8 gathers most solutions to the exercises of the previous chapters.

## 数学代写|代数拓扑代写Algebraic topology代考|Homology and cohomology theories

• 对于连续的地图 $f: X \rightarrow Y$ 和 $g: Y \rightarrow Z$ ，我们有: $H_n(g f)=H_n(g) H_n(f)$
• 一个空间 $X$ 我们有: $H_n(\mathrm{id} X)=\operatorname{ID} H_n(X)$.
在范畴论的基本术语中，每个 $H_n$ 是协变函子，从拓扑空间（和连续映射）的类别 Top 到交换群 (和同态) 的类别 $A b$ 。作为一个明显的结果，每个 $H_n$ 保留可逆箭头：它将空间的同构变换为群 的同构。
事实上，同源理论以更丰富的形式出现：对于每一对“亲缘关系” $(X, A)$ 拓扑空间（其中 $A$ 是一个子空 间 $X$ )，和固定阿贝尔群 $G$ ，我们有一系列的组 $H_n(X, A ; G)$ 的相对同源性，系数在 $G$. 基本情况是具 有积分系数的绝对同调: $H_n(X)=H_n(X, \emptyset ; \mathbb{Z})$.
另一方面，上同调理论 $H^n(-)$ 采用连续映射 $f: X \rightarrow Y$ 同态 $H^n(f): H^n(Y) \rightarrow H^n(X)$ ，反转 箭头的方向，反转构图并保留身份。这里我们还有一个丰富的版本，用于拓扑空间的相对对和系数组 $G$
一个重要的事实出现了：系数在环中的上同调群 $R$ 有一个分级乘法，他们的家庭形成一个分级 $R$-代 数。我们将研究奇异上同调（奇异同调的副产品），并简要回顾其他理论，即 Alexander-Spanier 上 同调和 de Rham 上同调，它们自然以逆变形式出现。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。