# Revised Syllabus Introductory Topology II Mathematics 442 — Spring 2020

### Prof. Weibel (640:442)

• Lectures: TF2 (10:20-11:40AM) in ARC 207
• Text: Alan Hatcher; Algebraic Topology (free on-line edition); or Cambridge Univ. Press, 2001 (544pp.); (ISBN: 0-13-181629-2)

Catalogue listing Basic concepts of algebraic topology, including the fundamental group, plane curves, and a brief introduction to homology.

Prerequisites: 01:640:441 or permission of department.

• Weibel's Office hours
• Homework: Homework problems are listed in the schedule of lectures below.
Before Spring Break, they are due the following class.
After Spring Break, they are not to be turned in; solutions will be provided later.
I will be happy to answer homework-related questions anytime.

#### Revised Course Syllabus

Week Lecture dates Material HW assignment given (due Tuesday)
1 1/21, 24CW complexes Show that SX is a CW complex if X is; #0.2, #0.10
2 2/4, 7The fundamental group Ch.1.1 #6,12,13,16(a,b,c),19
3 2/11, 14Van Kampen's Theorem Ch.1.2 #6,7,11,15
4 2/18, 21Covering spaces Ch.1.3 #1,4,9,14
5 2/25, 28Axioms for homology Compute H* for the torus and RP2 over F=Z/2, F=R
6 3/3, 6Simplicial homology Ch.2.2 #4,7,12;   2.3 #3,4
7 3/10review, midterm midterm on Tuesday 3/10
8 3/13, 17, 20Spring Break (no class) move class to on-line
9 3/24, 27Simplicial complexes Ch.2.1 #1,2,8,17 Solutions
10 3/31, 4/3Singular homology Ch.2.1 #15,21,24,29 Solutions
11 4/7, 4/10Cohomology Ch.3.1 #8(a,c),9,12,13 (take G=Z/2) Solutions
12 4/14, 4/17Cohomology 2B#1,  Ch.3,2 #3a,7
Prove the Jordan Curve Theorem Solutions
13 4/21, 4/24Vector bundles 1) if n is odd, show that the tangent bundle T to Sn has a nowhere-zero section
2) If E→X has patching maps gij, show that the patching maps det(gij) define a line bundle
3) Show that a map f: X→ Grn detemines a vector bundle on X.
Solutions
14 4/28, 5/1Vector Bundles and K-theory 1) Show that the clutching map for the tangent bundle on the 2-sphere has degree 2
2) Compute KO(Sn) for n=0,1,2,3
3) Compute K(X ∨ Y ∨ Z), if X,Y,X are connected
Solutions
155/4 (Monday)Homotopy theory Review for Final
16 May 13
(Wednesday)
8-11 AM Final Exam (cumulative)

• Assessment: Grades will be determined by:
the midterm (40%), the Final Exam (40%),
homework before Spring Break (10%), and class participation (10%).

Charles Weibel / weibel@math.rutgers.edu / Spring 2020