Chad Giusti
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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

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
Reading:  Section 1.1
Prelim Ex: S1.1:  1, 2, 7
HW 
Feb 19
Categories and functors
The fundamental group
Reading:  Section 1.2
Prelim Ex
HW
Feb 26
Covering spaces (pt 1) 
Homotopy Lifting Property
Reading: Section 1.3, p. 56-62
Prelim Ex: S1.3: 2, 3
HW
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
 
 
  
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:
May 7
 
Final Exam out (Due 5/24 at midnight)
Final Exam
May 14
 
NO CLASS
N/A













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