Math 6452

Differential Topology

Fall 2019


Class Meetings

Monday, Wednesday, Friday, 10:10-11:00 am, Skiles 308.


Office Hours

In Skiles 234, Monday 3-4, Wednesday 2-3, and by appointment


There will be daily homework assignments, each with 1-3 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.


There will be one midterm exam (Oct 11) and a final exam (Dec 6).


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

  1. For a set M, say that an n-dimensional atlas for M is (1) a collection of subsets {Ui} of M whose union is M, (2) a collection of open sets Vi of Rn, and (3) a collection of bijective functions φi : Ui → Vi with the properties that (i) each φi(Ui ∩ Uj) is open and (ii) for all Ui ∩ Uj the maps φjφi-1 : φi(Ui ∩ Uj) → Rn are continuous. Show that a set M with such an n-dimensional atlas has a topology where the open sets are the subsets U of M with the property that each φi(U ∩ U_i) is open. Show that if this topology is second countable and Hausdorff, then M with this topology is an n-manifold (meaning, a second countable, Hausdorff topological space that is locally homeomorphic to Rn), and that every n-manifold has such an atlas.
  2. Read about group actions and the quotient topology.
  3. 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.
  4. Show that the manifold MB = [0,1] x R/~ with (0,x)~(1,-x) is a smooth vector bundle over S1. Show that it is not isomorphic to the trivial bundle.
  5. Show that the tautological bundle over RP1 is isomorphic to the nontrivial line bundle over the circle in the previous exercise.
  6. Show that the tangent bundle on the torus is isomorphic to the trivial bundle.
  7. 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.
  8. Prove the Jacobi identity for the Lie bracket on vector fields.


---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