# Final Exam

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Fall
1 Sep 12 About, Tue, Thu, Std2Disc
2 Sep 19 Tue, Thu, HW1, 14 Sets
3 Sep 26 Tue, Thu, Photo
4 Oct 3 Tue, Thu
5 Oct 10 HW2, Tue, Thu
6 Oct 17 Tue, Thu, HW3
7 Oct 24 Mon, Tue, Thu
8 Oct 31 Tue, Thu, HW4
9 Nov 7 TE1, Tue, Thu
10 Nov 14 Tue, Thu, HW5
11 Nov 21 Tue, Thu
12 Nov 28 Tue, Thu, HW6
13 Dec 5 Tue, Thu
E Dec 12 TE2
Spring
14 Jan 9 Tue, IT83, Thu, HW7
15 Jan 16 Tue, Thu
16 Jan 23 Tue, HW8, Thu
17 Jan 30 Tue, Thu
18 Feb 6 TE3, Tue, Thu
19 Feb 13 Tue, Thu
R Feb 20
20 Feb 27 Tue, Thu, HW9
21 Mar 6 Tue, Thu, HW10
22 Mar 13 Tue, Thu
23 Mar 20 Tue, Thu, HW11
24 Mar 27 Tue, Thu
25 Apr 3 Tue, Thu, HW12
26 Apr 10 Tue, Thu
Study Apr 17 Office Hours
Exams Apr 24 Final, PM

The Final Exam will take place on Monday April 24 from 2PM until 5PM at Wilson Hall (New College) 1016 (http://www.osm.utoronto.ca/cgi-bin/class_spec/spec03?bldg=WI&room=1016). On Thursday April 20 and on Friday April 21 between 11-2 (watch here for changes!) I will be in my office with my door open, available to answer questions. Though if a group of students will hang out at the math lounge, so will I. (Allow me a quick lunch break).

Most likely, the exam will have two parts; Part I will consist of 3 questions on first semester material (all included) and Part II will have 6 questions on second semester material. Math 427S students will be required to solve 5 questions in part II and nothing from part I. Math 1300Y students (and students taking this exam as a comprehensive exam) will be required to solve 4 questions in part II and 2 questions in Part I. (So Math 1300Y will have to work harder; it is unfair but yet, they get a full year credit for the course rather than just half-year).

Sample exams are available on last year's web site (http://www.math.toronto.edu/~drorbn/classes/0405/Topology/index.html).

The material is hard but beautiful; start early so you can study with love rather than pressure.

Following the final we will hold a class Post Mortem discussion.

Good Luck!!

The Final Exam indeed took place as scheduled. The average grade for 1300 (427) was 84.09 (73.83), the median 80.5 (76.5) and the standard deviation was 13.98 (17.93). The overall class grade (with all other factors counted) average was 88.59 (74.33), median 89.5 (77.5) and standard deviation 8.85 (16.74). Here it is:

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!

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!

Duration. You have 3 hours to write this exam.

Allowed Material. None.

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 connected, so is $X\times X$.

Problem 2. Let X be a topological space, let C(X,I) be the set of all continuous functions on X with values in the unit interval I = [0,1], and let $\iota:X\to I^{C(X,I)}$ be defined by ι(x)f: = f(x) for $x\in X$ and for $f\in C(X,I)$.

1. Prove that ι is continuos.
2. Define the phrase "X is completely regular (T3.5)".
3. Prove that if X is completely regular then ι is one to one.
4. Prove that if X is completely regular then ι is a homeomorphism of X to ι(X).

Problem 3.

1. Define the phrase "G is a topological group".
2. Prove that if G is a topological group then π1(G) is Abelian.

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) = eit. Is it true that every map $f:{\mathbb R}P^2\to S^1$ can be lifted to a map $\tilde{f}:{\mathbb R}P^2\to{\mathbb R}$ such that $f=p\circ\tilde{f}$? Justify your answer.

Problem 5. Let X be a topological space and let A be a subspace of X that has a neighborhood V that deformation retracts back to A. Prove that $\tilde{H}_\star(X/A)\simeq H_\star(X,A)$. Feel free to use excision, the exact sequences associated with pairs and triples of topological spaces and/or any other result proven in class prior to this particular isomorphism.

Problem 6. A Δ-space X is given by $S_2=\{A,B\}\longrightarrow S_1=\{a,b,c\} \longrightarrow S_0=\{P,Q\}$, where $\partial_{0,1,2}(A)=(c,b,a)$, $\partial_{0,1,2}(B)=(c,a,b)$, $\partial_{0,1}(a)=(Q,P)$, $\partial_{0,1}(b)=(Q,P)$ and $\partial_{0,1}(c)=(Q,Q)$.

1. Write down the complex $C^\Delta_\star(X)$ (including the boundary maps).
2. Calculate the homology of X. (I.e., calculate $H^\Delta_k(X)$ for all k).
3. Can you identify the topological space | X | ?

Problem 7. Let γ be an embedding of the "figure 8 space" $S^1\vee S^1$ into ${\mathbb R}^3$. Prove that $H_1({\mathbb R}^3-\gamma(S^1\vee S^1))\simeq{\mathbb Z}\oplus{\mathbb Z}$.

Problem 8. Let $f:S^n\to S^n$ be a continous map and let $y\in S^n$ be so that f − 1(y) is finite.

1. Define "the degree deg(f) of f".
2. For $x\in f^{-1}(y)$, define "the local degree degx(f) of f at x".
3. What is the "local degree formula"?
4. If it is also given that f is even (i.e., f(x) = f( − x) for all $x\in S^n$), show that deg(f) is also even. Be careful to separate the cases where n is even and where n is odd.

Problem 9. Let A, B and C be chain complexes, let f1 and f2 be chain-complex morphisms from A to B and let g1 and g2 be chain-complex morphisms from B to C.

1. Define "f1 is homotopic to f2".
2. Prove that if f1 is homotopic to f2 and if g1 is homotopic to g2, then $g_1\circ f_1$ is homotopic to $g_2\circ f_2$.

Good Luck!