MTH 532/632: Differential Topology

University of Miami, Spring 2026

Instructor: Christopher Scaduto
Email: c.scaduto @ math.miami.edu
Office: Ungar 525
Office hours: 10am-11am Tuesday, 8am-9am Thursday, or by appointment
Time and Location: TuTh 11am-12:15pm, Dooly Memorial 209

Course syllabus can be found here.

Text: Differential Topology by Guillemin and Pollack

The symbol § used below means "Section".

Homework Assignments

All homeworks are worth the same, even if graded out of different totals.

Assignment	Due Date	Remarks
Homework 1: Ch 1 § 1: # 2, 3, 7, 9, 12; Ch 1 § 2: # 2, 4, 8	1/22/26
Homework 2: Ch 1 § 3: # 1, 2; Ch 1 § 4: # 1, 2, 5, 6, 7	2/3/26
Homework 3: Ch 1 § 4: # 11, 12; Ch 1 § 7: # 1, 3, 4, 7	2/13/26
Homework 4: Ch 1 § 7: # 8; Ch 2 § 1: # 4, 5, 9; Ch 2 § 2: # 3, 6, 7	3/5/26
Homework 5: Ch 1 § 5: # 1, 2, 4, 6, 7, 9; Ch 2 § 3: # 4	3/24/26
Homework 6: Ch 2 § 4: # 4, 5, 6, 7, 10, 11; Ch 3 § 2: # 2, 4	4/9/26

Course Schedule

Date	Lecture Content	Reading
1/13/26	Syllabus overview, introduction, definition of smooth manifold, examples	§ 1.1
1/15/26	Survey on classification of manifolds, derivatives and tangent spaces	§ 1.2
1/20/26	Class cancelled by Provost
1/22/26	More derivatives. Examples: sphere, Hopf map.	§ 1.3, 1.4
1/27/26	Local structure of smooth maps. Rank Theorem.	§ 1.3, 1.4
1/29/26	Critical/regular values. Preimage Theorem.	§ 1.4
2/3/26	Proof and applications of the Preimage Theorem.	§ 1.4
2/5/26	Orthogonal group. Submanifolds and embeddings.	§ 1.4, § 1.3
2/10/26	Sard's Theorem. Sets of measure zero. Statement of (weak) Whitney embedding theorem.	§ 1.7
2/12/26	Fiber bundles and vector bundles. Proof of (weak) Whitney embedding theorem.	§ 1.8
2/17/26	Sard's Theorem Application: Homotopy groups of spheres. Manifolds with boundary.	§ 2.1
2/19/26	Preimage Theorem V.2. Retraction theorem. Brouwer fixed point theorem.	§ 2.2
2/24/26	Intro to Morse functions. Abstract manifolds.	§ 1.7
2/26/26	Exam
3/3/26	Embedding abstract manifolds into Euclidean space. Real projective space.
3/5/26	Transversality. The Transversality Theorem.	§ 2.3
3/10/26	Spring break
3/12/26	Spring break
3/17/26	Intersection Theory Mod 2.	§ 2.4
3/19/26	Intersection numbers for more than 2 submanifolds. Application to coverings by open balls.
3/24/26	Fundamental Thm of Algebra. Jordan-Brouwer and Whitney-Graustein Theorems.	§ 2.4, 2.5
3/26/26	Orientations of vector spaces and manifolds.	§ 3.2
3/31/26	Class cancelled
4/2/26	Orientations on manifolds with boundary. Hypersurfaces. Oriented intersection numbers.	§ 3.2
4/7/26
4/9/26
4/14/26
4/16/26
4/21/26
4/23/26