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.

1.4 (M) Suppose $\Gamma$ is a connected graph where each node has even degree. Prove that every graph obtained by removing one edge from $\Gamma$ is still connected.

1.5 Let $v_1, \ldots, v_n$ be the nodes of a graph $\Gamma$ and $A$ the adjacency matrix: this is the $n \times n$-matrix whose entry $a_{i j}$ equals the number of edges joining $v_i$ to $v_j$. Explain the geometrical meaning of the entries of the powers of $A$.

问题 2.

1.6 Tralfamadore’s zoo has six thematic areas surrounded by walking paths, as in Fig. 1.4. Altogether there are 16 pathways that are long $1 \mathrm{~km}$ and two that are long $2 \mathrm{~km}$. Once appointed director of the zoo, Mister V. realised that the zoo’s janitor was going around $24 \mathrm{~km}$ every day to keep sidewalks tidy, and blamed him for wasting time. The caretaker said that to sweep every path, starting from his shed and then making a way back, he really needed to go around as much. Who’s right according to you?

1.7 (P) Say whether Fig. 1.6-after glueing sides as indicated-will produce a disc, a cylinder, a Möbius strip or something completely different. Explain your answer.

问题 3.

1.8 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.

问题 4.

1.9 Which of the following sets are closed in $\mathbb{R}^2$ ?
$$
\begin{array}{cc}
\left{(x, y) \mid x^2+2 y^2=1\right}, & {(x, y) \mid 0 \leq x \leq 1,0 \leq y<1}, \
{(x, y) \mid 0 \leq x, 0 \leq y}, & {(x, y) \mid 0 \leq x \leq 1, x+y \leq 1}, \
\left{(x, y) \mid 0<x^2+y^2 \leq 1\right}, & {(x, y) \mid 0 \leq x, 0 \leq y \leq \sin (x)} .
\end{array}
$$
1.10 Establish which of the following sets are closed in $\mathbb{C}(i=\sqrt{-1})$ :
$$
\begin{aligned}
& \left{z \in \mathbb{C} \mid z^2 \in \mathbb{R}\right}, \quad\left{z \in \mathbb{C}|| z^2-z \mid \leq 1\right}, \
& \left{2^n+i 2^n \mid n \in \mathbb{Z}\right}, \quad\left{2^{-n}+i 2^n \mid n \in \mathbb{Z}\right} .
\end{aligned}
$$