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A Sample Final Exam

University of Toronto, April 12, 2005

This document in PDF: SampleFinal.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 ``product topology'' on $X\times X$.
2. Prove that if $X$ is compact then so is $X\times X$.
3. Prove that the ``folding of $X$ along the diagonal'', $S^2X:=X\times X/(x,y)\sim(y,x)$ is also compact.

Problem 2. Let $X$ be a compact metric space and let $\{U_\alpha\mid\alpha\in A\}$ be an open cover of $X$. Show that there exists $\epsilon>0$ such that for every $x\in X$ there exists $\alpha\in A$ such that the $\epsilon$-ball centred at $x$ is contained in $U_\alpha$. ($\epsilon$ is called a Lebesgue number for the covering.)

Problem 3.

1. Compute $\pi_1({\mathbb{R}}{\mathbb{P}}^2)$.
2. A topological space $X_f$ is obtained from a topological space $X$ by gluing to $X$ an $n$-dimensional cell $e^n$ using a continuous gluing map $f:\partial e^n=S^{n-1}\to X$, where $n\geq 3$. Prove that obvious map $\iota:\pi_1(X)\to\pi_1(X_f)$ is an isomorphism.
3. Compute $\pi_1({\mathbb{R}}{\mathbb{P}}^n)$ for all $n$.

Part B

Problem 4. Let $p:X\to B$ be a covering of a connected locally connected and semi-locally simply connected base $B$ with basepoint $b$. Prove that if $p_\star\pi_1(X)$ is normal in $\pi_1(B)$ then the group of automorphisms of $X$ acts transitively on $p^{-1}(b)$.

Problem 5. A topological space $X_f$ is obtained from a topological space $X$ by gluing to $X$ an $n$-dimensional cell $e^n$ using a continuous gluing map $f:\partial e^n=S^{n-1}\to X$, where $n\geq 2$. Show that

1. $H_m(X)\cong H_m (X_f)$ for $m\ne n,n-1$.
2. There is an exact sequence
   $$0 \to H_n(X) \to H_n(X_f) \to H_{n-1}(S^{n-1}) \to\ H_{n-1} (X) \to H_{n-1}(X_f) \to 0.$$

Problem 6. Let $T$ denote the (standard) 2-dimensional torus.

1. State the homology and cohomology of $T$ including the ring structure. (Just state the results; no justification is required.)
2. Show that every map $f$ from the sphere $S^2$ to $T$ induces the zero map on cohomology. (Hint: cohomology flows against the direction of $f$).

Problem 7. For $n\geq 1$, what is the degree of the antipodal map on $S^n$? Give an example of a continuous map $f:S^n \rightarrow S^n$ of degree 2 (your exmple should work for every $n$). Explain your answers.

Problem 8.

1. State the ``Salad Bowl Theorem''.
2. State the ``Borsuk-Ulam Theorem''.
3. Prove that the latter implies the former.

Problem 9. Suppose

$$\xymatrix{ A\ar[r]^{a}\ar[d]^\alpha & B\ar[r]^{b}\ar[d]^\beta & C... ...psilon \\ A'\ar[r]^{a'} & B'\ar[r]^{b'} & C'\ar[r]^{c'} & D'\ar[r]^{d'} & E' }$$

is a commutative diagram of Abelian groups in which the rows are exact and $\alpha$, $\beta$, $\delta$ and $\epsilon$ are isomorphisms. Prove that $\gamma$ is also an isomorphism.

Good Luck!

Warning: The real exam will be similar to this sample, to my taste. Your taste may be significantly different.

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