MTH 532/632: Topology II (Differential Topology)

University of Miami, Spring 2025

Instructor: Christopher Scaduto
Email: c.scaduto @ math.miami.edu
Office: Ungar 525
Office hours: 10-11 Tuesday, or by appointment

Class Time and Location: 11:00-12:15 Tuesdays and Thursdays in Dooly Memorial 112.

Course syllabus can be found here.

Text: Differential Topology by Guillemin and Pollack

The symbol § used below means "Section".

Homework Assignments



Assignment Due Date
Homework 1: Ch 1 § 1: # 2, 3, 7, 9, 12; Ch 1 § 2: # 2, 4, 8 1/23/25
Homework 2: Ch 1 § 3: # 1, 2; Ch 1 § 4: # 1, 2, 5, 6, 7 2/4/25
Homework 3: Ch 1 § 4: # 11, 12; Ch 1 § 7: # 1, 3, 4, 7 2/13/25
Homework 4: Ch 1 § 7: # 8; Ch 2 § 1: # 4, 5, 9; Ch 2 § 2: # 3, 6, 7 3/4/25
Homework 5: Ch 1 § 5: # 1, 2, 4, 6, 7, 9 3/20/25
Homework 6: Ch 2 § 3: # 4; Ch 2 § 4: # 4, 5, 6, 7, 10, 11 3/28/25
Homework 7: Ch 4 § 2: # 1, 2, 5; Ch 4 § 3: # 5, 7, 8, 9, 10, 12; Ch 4 § 5: # 1 4/22/25


Course Schedule

Date Lecture Content Reading Notes
1/14/23 Introduction. Smooth functions. Definition of manifolds, examples. § 1.1
1/16/23 More examples of manifolds. Tangent spaces and derivatives. § 1.1, 1.2
1/21/23 More on derivatives. Chain rule. Example: spheres. § 1.2
1/23/23 Detailed examples: Hopf map, 2-torus. § 1.3, 1.4
1/28/23 Local structure of smooth maps. Rank Theorem. § 1.3, 1.4
1/30/23 Critical/regular values. Preimage Theorem. § 1.4
2/4/23 Applications of Preimage Theorem. Submanifolds and embeddings. § 1.4, § 1.3
2/6/23 More on embeddings. Sard's Theorem. § 1.3, § 1.7
2/11/23 Fiber and vector bundles. Tangent bundles. Whitney's (weak) Embedding Theorem. § 1.8
2/13/23 Sard's Theorem Application: Homotopy groups of spheres. Manifolds with boundary. § 2.1
2/18/23 Manifolds with boundary (cont.). Preimage Theorem V.2. Classification of 1-manifolds. § 2.1, 2.2
2/20/23 Exam 1
2/25/23 Retraction theorem. Brouwer fixed point theorem. Intro to Morse functions. § 2.2
2/27/23 More Morse functions. Abstract manifolds. § 1.7
3/4/23 Real projective space. Embedding abstract manifolds.
3/6/23 Transversality. § 1.5, 2.3
3/11/23 Spring break
3/13/23 Spring break
3/18/23 Transverslity Theorem and related results. § 2.3
3/20/23 Intersection Theory mod 2. § 2.4
3/25/23 Intersection Theorem mod 2 continued. § 2.4
3/27/23 Intersection numbers for more than 2 submanifolds. Application to coverings by open balls.
4/1/23 Fundamental Thm of Algebra! Jordan-Brouwer and Whitney-Graustein Theorems. § 2.4, 2.5
4/3/23 Exam 2
4/8/23 Multilinear algebra. Alternating tensors. § 4.1, 4.2
4/10/23 More exterior algebra. Differential forms. § 4.2, 4.3
4/15/23
4/17/23
4/22/23
4/24/23