# Math 141

## UC Berkeley, Spring, 2010

## Instructor

Rob Kirby

*kirby@math.berkeley.edu*

Office phone: 510-642-0845.

Office: 919 Evans.

Office hours: 9-10am MF and by appointment.
## Lectures (MWF 8-9pm)

Lectures take place on Monday, Wednesday and Friday from 8:10am to 9:00am
in 5 Evans.
## Textbooks

The text for Math 141 is Differential Topology by Guillemin and Pollack.

## Exams

Final exam: 10 May, 8am, 3 Evans.
Midterm exam:

26 April, 2010

## Grades

The written midterm will count 100 points, the oral exam will count,
and the final exam will count 200 points.
## Homework assignments

Homework is normally due on Mondays.

The Problem

Let f:[0,1]-->[0,1] be the function defined to be 0 at the irrationals
and 1/q when x = p/q in reduced form.

1. For what points in x in [0,1] is f continuous? Prove your
answer to this question, and all the following also..

2. For what points in x in [0,1] is f differentiable?

3. Does the integral of f over [0,1] exist?

Let F:[0,1]x[0,1] --> [0,1] be defined by F(x,y) = f(x)f(y).

4. For what points (x,y) in [0,1]x[0,1] is F continuous?

5. Is F differentiable at (0,0)?

6. Does the integral of F over [0,1]x[0,1] exist?

Due this semester.

Page 5: 2, 4, 6, 18.

Page 12: 11, 12.

Due Monday, 25 January.

Page 18: 2, 9.

Page 25: 5, 7, 8, 10.

Due Monday 1 February.

Page 32: 2, 10, 11.

Page 37: 2, 3, 4, 6, 7.

Due Monday 8 February.

Page 45: 1. 4. 6. 9. 17.

Due Monday 22 February.

Page 76: 16.

Due Monday, 1 March.

A. If X and Y are CW complexes, show that XxY is a CW complex.

B. Calculate the homology groups of S^1 x S^2.

C. Calculate the homology groups of the 3-torus.

D. Calculate the homology groups of real projective n-space.

Due Monday, 8 March.

Page 160: 5, 10, 11.

Page 172: 5, 7.

Due Monday 12 April.

Page 172: 8, 9, 10, 12.

Page 185: 2, 8.

Due Monday 19 April.

Rob Kirby
Last modified: Wed Apr 14 14:10:15 PDT 2010