Math 6452 Home Page

Math 6452 Differential Topology Fall 2019 -

Class Meetings

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

Texts

Office Hours

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

Homework

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.

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 n-dimensional 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 Rⁿ, 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_jφ_i^-1 : φ_i(U_i ∩ U_j) → 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 φ_i(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.
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 S¹. Show that it is not isomorphic to the trivial bundle.
Show that the tautological bundle over RP¹ 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.