© | Dror Bar-Natan: Classes: 2004-05: Math 1300Y - Topology:

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Final Exam

University of Toronto, April 29, 2005

This document in PDF: Final.pdf

Math 1300Y Students: Make sure to write ``1300Y'' in the course field on the exam notebook. Solve 2 of the 3 problems in part A and 4 of the 6 problems in part B. Each problem is worth 17 points, to a maximal total grade of 102. If you solve more than the required 2 in 3 and 4 in 6, indicate very clearly which problems you want graded; otherwise random ones will be left out at grading and they may be your best ones! You have 3 hours. No outside material other than stationary is allowed.

Math 427S Students: Make sure to write ``427S'' in the course field on the exam notebook. Solve 5 of the 6 problems in part B, do not solve anything in part A. Each problem is worth 20 points. If you solve more than the required 5 in 6, indicate very clearly which problems you want graded; otherwise random ones will be left out at grading and they may be your best ones! You have 3 hours. No outside material other than stationary is allowed.

Good Luck!

Part A

Problem 1. Let $X$ be a topological space.

1. Define the phrase ``$X$ is Hausdorff''.
2. Define the phrase ``$X$ is normal''.
3. Define the phrase ``$X$ is compact''.
4. Prove that if $X$ is compact and Hausdorff, it is normal.

Problem 2. Let $X$ be a metric space.

1. Define the phrase ``$X$ is complete''.
2. Define the phrase ``$X$ is totally bounded''.
3. Prove that if $X$ is totally bounded and complete than every sequence in $X$ has a convergent subsequence.

Problem 3.

1. State the Van Kampen theorem in full.
2. Let $D=\{z\in{\mathbb{C}}:\vert z\vert\leq 1\}$ be the unit disk in the complex plane and let $Y$ be its quotient by the relation $z\sim ze^{2\pi i/3}$, for $\vert z\vert=1$. Compute $\pi_1(Y)$.

Part B

Problem 4.

1. Let $p:X\to B$ be covering map and let $f:Y\to B$ be a continuous map. State in full the lifting theorem, which gives necessary and sufficient conditions for the existence and uniqueness of a lift of $f$ to a map $\tilde{f}:Y\to X$ such that $f=p\circ\tilde{f}$.
2. Let $p:{\mathbb{R}}\to S^1$ be given by $p(t)=e^{it}$. Is it true that every map $f:{\mathbb{R}}{\mathbb{P}}^2\to S^1$ can be lifted to a map $\tilde{f}:{\mathbb{R}}{\mathbb{P}}^2\to{\mathbb{R}}$ such that $f=p\circ\tilde{f}$? Justify your answer.

Problem 5. Let $M$ be an $n$-dimensional topological manifold (a space in which every point has a neighborhood homeomorphic to ${\mathbb{R}}^n$), and let $p$ be a point in $M$.

1. Show that $p$ has a neighborhood $U$ for which $H_k(M-p,U-p)$ is isomorphic to $\tilde{H}_k(M)$ for all $k$, and so that $U$ is homeomorphic to a ball.
2. Write the long exact sequence corresponding to the pair $(M-p,U-p)$.
3. Prove that $\tilde{H}_k(M-p)$ is isomorphic to $\tilde{H}_k(M)$ for $k<n-1$.

Problem 6.

1. Present the space $X=S^2\times S^4$ as a CW complex.
2. Calculate the homology of $X$. (I.e., calculate $H_k(X)$ for all $k$).
3. What is the minimal number of cells required to present $X$ as a CW complex? Justify your answer.

Problem 7.

1. Define the degree $\deg\Phi$ of a continuous map $\Phi:T^2\to S^2$.
2. Let $\gamma_1,\gamma_2:S^1\to{\mathbb{R}}^3$ be two continuous maps such that $\gamma_1(S^1)\cap\gamma_2(S^1)=\emptyset$. Let $\Phi_{\gamma_1,\gamma_2}:T^2=S^1\times S^1\to S^2$ be defined by
   $$\Phi_{\gamma_1,\gamma_2}(z_1,z_2):= \frac {\gamma_2(z_2)-\gamma_1(z_1)} {\vert\gamma_2(z_2)-\gamma_1(z_1)\vert},$$
   for $z_1,z_2\in S^1$. Prove that the degree $l(\gamma_1,\gamma_2):=\deg\Phi_{\gamma_1,\gamma_2}$ is invariant under homotopies of $\gamma_1$ and $\gamma_2$ throughout which $\gamma_1$ and $\gamma_2$ remain disjoint. (I.e., homotopies $\gamma_{1,t}$ and $\gamma_{2,t}$ for which $\gamma_{1,t}(S^1)\cap\gamma_{2,t}(S^1)=\emptyset$ for all $t$).
3. Compute (without worrying about signs, but otherwise with justification) the degree $l(\gamma_1,\gamma_2)$ where $\gamma_1$ and $\gamma_2$ are given by the picture $\bigcirc\bigcirc$.
4. Compute (without worrying about signs, but otherwise with justification) the degree $l(\gamma_1,\gamma_2)$ where $\gamma_1$ and $\gamma_2$ are given by the picture $\HopfLink$.

Problem 8.

1. State the theorem about the homology of the complement of an embedded disk in ${\mathbb{R}}^n$.
2. State the theorem about the homology of the complement of an embedded sphere in ${\mathbb{R}}^n$.
3. Prove that the first of these two theorems implies the second.

Problem 9. A chain complex $A$ is said to be ``acyclic'' if its homology vanishes (i.e., if it is an exact sequence). Let $C$ be a subcomplex of some chain complex $B$.

1. Show that if $C$ is acyclic then the homology of $B$ is isomorphic to the homology of $B/C$ (so $C$ ``doesn't matter'').
2. Show that if $B/C$ is acyclic then the homology of $B$ is isomorphic to the homology of $C$ (so ``the part of $B$ out of $C$'' doesn't matter).
3. If $B$ is acyclic, can you say anything about the relation between the homology of $C$ and the homology of $B/C$?

Good Luck!

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