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 BorsukUlum theorem.
Course Metadata:
Class Spacetime Coordinates: MW 8:409:55 AM , Gore 219
Office Hours: M 1011 and 24, 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 takehome midterm exam, made available Wednesday, March 21 and due before class on Wednesday, April 4.
Final Exam: There will be a takehome 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 1011 and 24, 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 takehome midterm exam, made available Wednesday, March 21 and due before class on Wednesday, April 4.
Final Exam: There will be a takehome 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 2124 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 

Feb 26 
Covering spaces (pt 1) 
Homotopy Lifting Property 

Mar 5 
π_1(S^1) 
van Kampen's Theorem 
Reading: Section 1.3 p. 6368 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.6978 Prelim Ex: S1.3: 22, 31 HW: None this week 
Mar 19 

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 p102118 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 119131 Prelim Ex: S2.1: 12 HW: S2.1: 16, 18 
Apr 30 
Reading: Prelim Ex: HW: 

May 7 
Final Exam out (Due 5/24 at midnight) 
Final Exam 

May 14 
NO CLASS 
N/A 