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# 数学代写|代数拓扑代考Algebraic Topology代考|MA753 Cofibrations and Homotopy Pushouts

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## 数学代写|代数拓扑代考Algebraic Topology代考|Cofibrations and Homotopy Pushouts

A map $i: A \rightarrow X$ of spaces has the homotopy extension property (HEP) for a space $Y$ if for each homotopy $H: A \times I \rightarrow Y$ and for each map $f: X \rightarrow Y$ with $f(i(a))=H(a, 0)$ for all $a \in A$, there exists a homotopy $H^{\prime}: X \times I \rightarrow Y$ such that $H^{\prime}(i(a), t)=H(a, t)$ and $H^{\prime}(x, 0)=f(x)$ for all $a \in A, x \in X$, and $t \in I$. The homotopy $H^{\prime}$ is called an extension of $H$ with initial condition $f$. A map $i: A \rightarrow X$ is called a cofibration if it has the HEP for all spaces $Y$. Setting $i_0^A(a)=(a, 0)$ and $i_0^X(x)=(x, 0)$, we can express the definition by the diagram in which we require only existence of $H^{\prime}$, not uniqueness. The definition of cofibration is of course impractical to verify directly. Therefore it is good to know that if $i$ has the HEP for its own mapping cylinder $M_i$, then $i$ has the HEP for all spaces. To prove this, we first observe that the pushout diagram uniquely defines the map $s: M_i \rightarrow X \times I$. If $i$ is the inclusion of a subspace, then $s$ is a continuous bijection onto the image $X \times{0} \cup A \times I$. But the topology of $M_i$ might be finer than the subspace topology within $X \times I$. However, we see that $s$ is a homeomorphism onto the image, hence a subspace inclusion, if $i$ and thus also $i \times$ id is the inclusion of a closed subspace. By Proposition $2.12$ (iii) below, $s$ is also a homeomorphism onto the image if $i$ is a cofibration.

## 数学代写|代数拓扑代考Algebraic Topology代考|Higher Homotopy Groups

As a reward for the hard work of the previous section, we obtain the following version of van Kampen’s theorem, which is a powerful tool to carry out actual computations of fundamental groups.

The fundamental group of a pointed space
$$\pi_1\left(X, x_0\right)=\left{\gamma:(I,{0,1}) \rightarrow\left(X, x_0\right)\right} / \simeq$$
is defined in terms of pointed homotopy classes of one-dimensional loops and hence encodes primarily low-dimensional data. It is therefore good at distinguishing lowdimensional spaces, for example we have
$$\pi_1\left(S^1, \bullet\right) \not \pi_1\left(S^2, \bullet\right)$$

## 数学代写|代数拓扑代考Algebraic Topology代考|Higher Homotopy Groups

$\backslash$ left 缺少或无法识别的分隔符

$$\pi_1\left(S^1, \bullet\right) \pi_1\left(S^2, \bullet\right)$$

## 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。