# A Sample Term Exam 2

University of Toronto, March 3, 2005

This document in PDF: SampleTE2.pdf

Math 1300Y Students: Make sure to write 1300Y'' in the course field on the exam notebook. Solve one of the two problems in part A and three of the four problems in part B. Each problem is worth 25 points. If you solve more than the required 1 in 2 and 3 in 4, 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 an hour and 50 minutes. 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 the four problems in part B, do not solve anything in part A. Each problem is worth 25 points. You have an hour and 50 minutes. No outside material other than stationary is allowed.

Part A

Problem 1. Consider .

1. Use to define the degree of a map .
2. A map is even if for all . Show that if is even then is even.
3. A map is odd if for all . Show that if is odd then is odd.

Problem 2. State Van-Kampen's theorem and compute the fundamental group of the Klein bottle (a square with two pairs opposite edges edges identified, one pair in a parallel manner and one pair in an anti-parallel manner).

Part B

Problem 3. Let be a connected, locally connected and semi-locally simply connected topological space with basepoint .

1. Explicitly construct the universal covering of '' as a set with a map and a basepoint .
2. Explicitly describe the topology of . You don't need to show that the topology you have described is indeed a topology or that is a covering map, or even that is continuous.
3. Without referring to the general classification of covering spaces show that if is a connected covering of (with basepoint ) then there is a unique basepoint-preserving such that .

Problem 4.

1. Define a morphism between two chain complexes''.
2. Show that a morphism between two chain complexes induces a map between their homologies.
3. Define a homotopy between two morphisms of chain complexes''.
4. Show that if and are homotopic morphisms of chain complexes, then .

Problem 5. A chain complex is said to be acyclic'' if its homology vanishes (i.e., if it is an exact sequence). Let be a subcomplex of some chain complex .

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

Problem 6. Let be a wedge of 5 lines, and , let be the result of gluing the ends of to each other with a twist, , with . The boundary'' of is a single circle that wraps around five times''. Let be the result of identifying that circle with the boundary of some disk .

1. Describe as a polygon with some edges identified.
2. Describe as the geometric realization of some -complex.
3. Compute the homology of .

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

The generation of this document was assisted by LATEX2HTML.

Dror Bar-Natan 2005-03-02