Course summary:
This course is an introduction to algebraic invariants of topological spaces. We will cover the construction of the fundamental group, higher homotopy groups, homology groups and cohomology rings of topological spaces. We will develop the languages of categories and homological algebra as a framework for these invariants, and apply a range of various geometric, algebraic and combinatorial techniques to compute topological invariants for spheres, Grassman manifolds, configuration spaces, and matrix Lie groups. We will prove classical topological results including cellular approximation, Poincare duality, fixed point theorems, the generalized Jordan curve theorem, and the Borsuk-Ulum theorem.
Course Metadata:
Class Spacetime Coordinates: MW 8:40-9:55 AM , Gore 219
Office Hours: M 10-11 and 2-4, Ewing 532, or by appointment
Texts: Algebraic Topology by Allen Hatcher
Deliverables: Readings, preliminary questions and homework will be posted to this web site on Wednesdays. Reading questions and solutions to preliminary questions are due the following Sundays at noon, and homework is due the following Wednesday by the beginning of class. All deliverables must be submitted electronically.
Midterms: There will be one take-home midterm exam, made available Wednesday, March 21 and due before class on Wednesday, April 4.
Final Exam: There will be a take-home midterm exam, made available Wednesday, May 9 and due before by midnight on Wednesday, May 23rd.
Grade Breakdown: 20% reading questions and preliminary exercises, 30% homework, 20% for the midterm exam, and 30% for the final exam.
Details on all of this and more: Course Syllabus
Office Hours: M 10-11 and 2-4, Ewing 532, or by appointment
Texts: Algebraic Topology by Allen Hatcher
Deliverables: Readings, preliminary questions and homework will be posted to this web site on Wednesdays. Reading questions and solutions to preliminary questions are due the following Sundays at noon, and homework is due the following Wednesday by the beginning of class. All deliverables must be submitted electronically.
Midterms: There will be one take-home midterm exam, made available Wednesday, March 21 and due before class on Wednesday, April 4.
Final Exam: There will be a take-home midterm exam, made available Wednesday, May 9 and due before by midnight on Wednesday, May 23rd.
Grade Breakdown: 20% reading questions and preliminary exercises, 30% homework, 20% for the midterm exam, and 30% for the final exam.
Details on all of this and more: Course Syllabus
Schedule (imagined by an optimist, subject to change) and deliverables:
Week |
Monday |
Wednesday |
Deliverables |
Feb 5 |
Introduction and "review" |
Quotient spaces and CW complexes |
Reading: Ch 0 + p 21-24 Prelim Ex: Ch 0, 3(b), 8, 14, 15 HW: none this week |
Feb 12 |
Homotopy |
Paths and \pi_0 |
|
Feb 19 |
Categories and functors |
The fundamental group |
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Feb 26 |
Covering spaces (pt 1) |
Homotopy Lifting Property |
|
Mar 5 |
π_1(S^1) |
van Kampen's Theorem |
Reading: Section 1.3 p. 63-68 Prelim Ex: S1.3: 6, 14 HW: S1.2: 2, 3, 8, 14 |
Mar 12 |
Lifting properties |
Classification of covering spaces |
Reading: Section 1.3 p.69-78 Prelim Ex: S1.3: 22, 31 HW: None this week |
Mar 19 |
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Mar 26 |
NO CLASS |
NO CLASS |
N/A |
Apr 2 |
Homotopy groups |
Exact sequences |
|
Apr 9 |
Homotopy LES of a pair |
Whitehead's Thm + Freudenthal Suspension Thm |
Reading: Ch 4, up to, but not including , the Hurewicz Thm. HW: S4.1: 5, 6, 12, 18 |
Apr 16 |
Why homotopy is hard to compute |
Singular simplices and chains |
Reading: Section 2.1 p102-118 Prelim Ex: S2.1: 3 HW: S4.1: 7, 20, S4.2: 1 |
Apr 23 |
Homology as a composite functor |
Homotopy invariance of homology and the snake lemma |
Reading: Section 2.1 p 119-131 Prelim Ex: S2.1: 12 HW: S2.1: 16, 18 |
Apr 30 |
Reading: Prelim Ex: HW: |
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May 7 |
Final Exam out (Due 5/24 at midnight) |
Final Exam |
|
May 14 |
NO CLASS |
N/A |