Introductory Topology II

Mathematics 442 — Spring 2020

**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, 24 CW complexes Show that SX is a CW complex if X is; #0.2, #0.10 2 2/4, 7 The fundamental group Ch.1.1 #6,12,13,16(a,b,c),19 3 2/11, 14 Van Kampen's Theorem Ch.1.2 #6,7,11,15 4 2/18, 21 Covering spaces Ch.1.3 #1,4,9,14 5 2/25, 28 Axioms for homology Compute H _{*}for the torus and RP^{2}over F=**Z**/2, F=**R**6 3/3, 6 Simplicial homology Ch.2.2 #4,7,12; 2.3 #3,4 7 3/10 review, midterm midterm on Tuesday 3/10 8 3/13, 17, 20 Spring Break (no class) move class to on-line 9 3/24, 27 Simplicial complexes Ch.2.1 #1,2,8,17 Solutions 10 3/31, 4/3 Singular homology Ch.2.1 #15,21,24,29 Solutions 11 4/7, 4/10 Cohomology Ch.3.1 #8(a,c),9,12,13 (take G=Z/2) Solutions 12 4/14, 4/17 Cohomology 2B#1, Ch.3,2 #3a,7

Prove the Jordan Curve Theorem Solutions13 4/21, 4/24 Vector bundles 1) if n is odd, show that the tangent bundle T to S ^{n}has a nowhere-zero section

2) If E→X has patching maps g_{ij}, show that the patching maps det(g_{ij}) define a line bundle

3) Show that a map f: X→ Gr_{n}detemines a vector bundle on X.

Solutions14 4/28, 5/1 Vector Bundles and K-theory 1) Show that the clutching map for the tangent bundle on the 2-sphere has degree 2

2) Compute KO(S^{n}) for n=0,1,2,3

3) Compute K(X ∨ Y ∨ Z), if X,Y,X are connected

Solutions15 5/4 (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