数学代写|Math8306 Algebraic topology 👍 👍 - 👍代考代写：100%准时可靠您的作业代写专家

Textbook: Algebraic Topology, by Allen Hatcher. The textbook is available at the University bookstore, and also on reserve in the Mathematics Library. Our textbook is also available free online, at http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Prerequisites

Course Outline: This is a first course in algebraic topology. The fall semester we plan to cover Chapters 0 – 3 of the textbook. The main topics are: homotopy, fundamental group, covering spaces, homology, and cohomology.

Web page: http://www.math.umn.edu/~akhmedov/MATH8306.html.

Grading: The course grade will be based on seven homework assignments, in class midterm, and a comprehensive take-home final, with the following weights:
42% Homework25% Midterm (in class)33% Take Home Final

Exams: There will be a comprehensive take-home final examination (which will worth 33% of the final course grade) and in class midterm on October 26 (which will worth 25% of the final course grade). TAKE HOME FINAL
DUE: by noon Monday, December 21

Homework: There will be 7 homeworks in this course, each worth 6 points. Homework will be a fundamental part of this course, and will be worth 42 points (42% of the course grade). NO LATE HOMEWORK WILL BE ACCEPTED. The first homework assignment will be due on September 23. Please staple your homework before handing it in. If you have questions about the homework, it is best to ask during my office hours.

Math8306 Algebraic topology HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

Consider the 8 types of interval, where $a, b$ are real numbers with $a<b$ :

$] a, b[={x \in \mathbb{R} \mid a<x<b}$;
$[a, b[={x \in \mathbb{R} \mid a \leq x<b}$;
$] a, b]={x \in \mathbb{R} \mid a<x \leq b}$;
$[a, b]={x \in \mathbb{R} \mid a \leq x \leq b}$;
$]-\infty, a[={x \in \mathbb{R} \mid x<a}$;
$]-\infty, a]={x \in \mathbb{R} \mid x \leq a}$;
$] a,+\infty[={x \in \mathbb{R} \mid a<x}$;
$[a,+\infty[={x \in \mathbb{R} \mid a \leq x}$.
Say which ones are closed subsets in $\mathbb{R}$ according to Definition 1.5.

Suppose we have a continuous map $\Phi: \mathbb{R}^n \rightarrow \mathbb{R}$ such that:

$\Phi(t x)=t \Phi(x)$ for any $t \geq 0$;

there exist constants $m, M>0$ for which:
$$
m|x| \leq \Phi(x) \leq M|x|, \quad \text { for every } \quad x \in \mathbb{R}^n
$$
Then $f, g: \mathbb{R}^n \rightarrow \mathbb{R}^n$ :
$$
f(x)=x \frac{\Phi(x)}{|x|}, \quad g(y)=y \frac{|y|}{\Phi(y)},
$$
are continuous and one inverse of the other. They induce a homeomorphism from ${x \mid \Phi(x) \leq 1}$ to the unit ball ${y \mid|y| \leq 1}$.
As for any $x=\left(x_1, \ldots, x_n\right) \in \mathbb{R}^n$ we have
$$
\frac{|x|}{\sqrt{n}} \leq \max \left(\left|x_1\right|, \ldots,\left|x_n\right|\right) \leq|x|,
$$
if we consider the $\operatorname{map} \Phi(x)=\max \left(\left|x_i\right|\right)$ the previous construction gives a homeomorphism between the unit ball and the hypercube
$$
[-1,1]^n=\left{x \in \mathbb{R}^n \mid \max \left(\left|x_1\right|, \ldots,\left|x_n\right|\right) \leq 1\right}
$$

问题 2.

$1.26(\odot)$ Check that
$$
{z \in \mathbb{C} \mid z=a+i b, b>0} \rightarrow{z \in \mathbb{C}|| z \mid<1}, \quad z \mapsto \frac{z-i}{z+i}
$$
is a homeomorphism and describe the inverse.

The maps
$$
\begin{aligned}
&{z \in \mathbb{C} \mid z=a+i b, b>0} \rightarrow{z \in \mathbb{C}|| z \mid<1}, z \mapsto \frac{z-i}{z+i}, \ &{z \in \mathbb{C}|| z \mid<1} \rightarrow{z \in \mathbb{C} \mid z=a+i b, b>0}, \quad z \mapsto i \frac{1+z}{1-z},
\end{aligned}
$$
are continuous and inverse of one other.

问题 3.

2.4 (๑) Let $f: X \rightarrow Y$ be a map and $A, B$ subsets of $X$. Tell which of the following statements are always true, and which are not:

$f(A \cap B) \supset f(A) \cap f(B)$;
$f(A \cap B) \subset f(A) \cap f(B)$;
$f(X-A) \subset f(X)-f(A)$;
$f(X-A) \supset f(X)-f(A)$;
$f^{-1}(f(A)) \subset A$;
$f^{-1}(f(A)) \supset A$.

Formulas 2, 4, 6 are always true, while 1,3 and 5 are usually false: consider for example $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=x^2, A={x \leq 0}, B={x \geq 0}$

问题 4.

(๑) Define $3^{\sqrt{2}}$ using solely the following notions: powers with integer exponent, well-ordering principle and completeness of the reals.

For instance
$$
3^{\sqrt{2}}=\sup \left{x \in \mathbb{R} \mid x^n \leq 3^{\min \left{m \in \mathbb{N} \mid m^2 \geq 2 n^2\right}} \text { for any } n \in \mathbb{N}\right}
$$