Math 6452
Differential Topology
Fall 2019




Class Meetings
Monday, Wednesday, Friday, 10:1011:00 am, Skiles 308.
Texts
Office Hours
In Skiles 234, Monday 34, Wednesday 23, and by appointment
Homework
There will be daily homework assignments, each with 13 problems. Homework is due at the start of the next class. The three lowest scores will be dropped. Homework will be posted on this web site. If there is no homework, the web site will explicitly say so. Homework may be due during the final instructional days.
Exams
There will be one midterm exam (Oct 11) and a final exam (Dec 6).
Grading
Course grades will be determined by homework scores and exam scores.
Weekly Schedule
Week 
Dates 
Topics 
Text Sections 
Homework 
Notes 
Comments 
1 
Aug 19 
Intro / Manifolds 

none 
none 
W p. 15 #2, A #1 
Intro 

2 
Aug 26 
Tangent spaces / Differentials 

W p. 30 #1,2,3 
W p. 38 #1,2,3,4 
A #2 


3 
Sep 2 
Quotients / IFT 

no class 
A.3 
W p. 48 # 1,2,4 

Labor day 
3 
Sep 9 
Regular level set thm 

8.1, 8.2, 8.3 
8.7, 8.8, 9.1 
9.5, 9.7, 11.1 


5 
Sep 16 
Transversality 

9.10 
none 
W p.66 # 1,2,3,8 


6 
Sep 23 
Vector bundles 

A.4, A.5, A.6 
A.7 
13.3 


7 
Sep 30 
Vector fields 

none 
14.2, 14.3, 14.5 
A.8, 14.8, 14.11 

Rosh Hashanah 
8 
Oct 7 
Lie bracket 

14.7, 14.12, 14.13 
none 
exam 

Yom Kippur 
9 
Oct 14 
Lie algebras 

no class 
15.3,15.5 
15.9, 15.10, 15.14 

Fall break 
10 
Lie group homomorphisms 


16.7, 16.8, 16.9 
16.11,16.12 
17.1 

Withdrawal deadline 
11 
Oct 28 
Differential forms 

17.2 
17.3, 17.4, 17.5 
18.5/td>
 

12 
Nov 4 
Exterior derivative 

18.3, 18.8, 18.9 
none 
19.1,19.2,19.3,20.9 


13 
Nov 11 
Orientation 

19.11,21.6,21.7,21.8,21.9 
22.8, 22.9, 22.10 
23.1 (don't hand in) 


14 
Nov 18 
Stokes 

23.3 
23.5, W 310.2, 310.3 
none 


15 
Nov 25 


none 
no class 
no class 

Thanksgiving 
16 
Dec 2 


none 
no class 
final 


Additional homework
 For a set M, say that an ndimensional atlas for M is (1) a collection of subsets {U_{i}} of M whose union is M, (2) a collection of open sets V_{i} of R^{n}, and (3) a collection of bijective functions φ_{i} : U_{i} → V_{i} with the properties that (i) each φ_{i}(U_{i} ∩ U_{j}) is open and (ii) for all U_{i} ∩ U_{j} the maps φ_{j}φ_{i}^{1} : φ_{i}(U_{i} ∩ U_{j}) → R^{n} are continuous. Show that a set M with such an ndimensional atlas has a topology where the open sets are the subsets U of M with the property that each φ_{i}(U ∩ U_i) is open. Show that if this topology is second countable and Hausdorff, then M with this topology is an nmanifold (meaning, a second countable, Hausdorff topological space that is locally homeomorphic to R^{n}), and that every nmanifold has such an atlas.
 Read about group actions and the quotient topology.
 Write up the proof that the quotient of a smooth manifold M by a free, properly discontinuous subgroup of Diff(M) is a manifold. If you want, you can assume the quotient is Hausdorff and second countable.
 Show that the manifold MB = [0,1] x R/~ with (0,x)~(1,x) is a smooth vector bundle over S^{1}. Show that it is not isomorphic to the trivial bundle.
 Show that the tautological bundle over RP^{1} is isomorphic to the nontrivial line bundle over the circle in the previous exercise.
 Show that the tangent bundle on the torus is isomorphic to the trivial bundle.
 Consider the bundle MB above. Show that every section of this bundle has a zero. In other words, MB fails to have a nonzero section.
 Prove the Jacobi identity for the Lie bracket on vector fields.
Resources
Calculus on Manifolds, Michael Spivak
Topology from the differentiable viewpoint, John W. Milnor
Differential topology (videos), John W. Milnor
Differential Topology, John W. Milnor and James Munkres
Differential Topology, Victor Guillemin and Alan Pollack
Guillemin and Pollack Errata, Theodore Shiffrin
Differential Topology, Bjørn Dundas
Textbooks in differential topology, Mladen Bestvina
Differentiable manifolds, Mladen Bestvina
Inverse Function Theorem
Sard's Theorem, Marco Gualtieri
Math 6510, Kevin Wortman