Introduction to Geometry and Topology

Course Description and Grading Breakdown
Math 109a is the first course in the Ma 109 sequence, Introduction to Geometry and Topology.  In the first part of the course, we will introduce notions of general point-set topology, basic examples and constructions.  Topics will include the notions of compactness, metrizability, separation properties, and completeness. The second part of the course will be an introduction to algebraic topology, including the fundamental group and homology theory.

Grading: HW 50%; Midterm 20%; Final 30%

Course Meeting Time and Location
Monday, Wednesday and Friday
10:00 - 10:55 am
B111 Downs Physics Laboratory (DWN)


Course Instructor Contact Information and Office Hours
411 Downs Physics Laboratory (DWN) 
office hours by appointment


TA Contact Information and Office Hours
TA:  Lingfei Yi;  email:  lyi@caltech.edu; Office: 210-C, Math Building(Building 15); Office hour: Friday 4-5pm

Course Schedule and Textbook

Textbooks: Munkres, "Topology", Pearson (whatever edition, it doesn't matter); Hatcher, "Algebraic Topology", Cambridge University Press (also freely available online): https://www.math.cornell.edu/~hatcher/AT/AT.pdf

The lectures may cover additional material that is not always contained in the textbooks, so attendance and participation in class is recommended.

 DateTopic 
 9-25set theory overview: boolean operations (rough notes: lecture 1)
 9-27
set theory overview: products, cardinality 
 9-29 set theory overview: infinite sets, axiom of choice, Zorn lemma  (rough notes: lectures 2 and 3)
 10-2 topologies: definition and basic properties (rough notes: lecture 4)
 10-4 closure, interior, induced topology (rough notes: lecture 5) 
 10-6continuity, metric spaces, product and quotient topology (rough notes: lecture 6)
 10-9 complete metric spaces (rough notes: lecture 6b)
 10-11 connectedness (rough notes: lecture 7)
 10-13 countability and separation 
 10-16 hausdorff spaces, compactness (rough notes: lecture 8)
 10-18 compactness in metric spaces (rough notes: lecture 9)
 10-20 compactness limit point, and sequential compactness
 10-23
 metrizability (rough notes: lecture 10)
 10-25 topological surfaces classification
 10-27 topological surfaces classification
 10-30 topological surfaces classification (rough notes lectures 11-12-13)
 11-1 triangulations of surfaces, Euler characteristic, homotopy (rough notes: addendum)
 11-3 triangulation of surfaces: the Jordan-Schoenflies theorem (rough notes)
 11-6 Peano curves; nowhere differentiable curves
 11-8 Fundamental group: definition and first properties
 11-10 triangulation of surfaces (continued) (rough notes)
 11-13 fundamental groups and covering spaces (rough notes)
 11-15 fundamental group, covering spaces and Seifert van Kampen theorem (rough notes)
 11-17 problem set discussion
 11-20 fundamental group, covering spaces and Seifert van Kampen, continued
 11-27 homology: chains, cycles, boundaries, chain complexes
 11-29simplicial homology groups, Euler characteristic revisited
 12-1 singular homology, Steenrod axioms, homotopy invariance
(rough notes: lecture 16) and (part b: remaining pages)

Note: No class on Wednesday November 22 (day before Thanksgiving)

Additional Reading Material



Homotopy groups of diagonal complements

Course Policies

Homework is due at 3PM on Mondays in the homework math drop boxes in Downs, near classroom 113 (directly across from the elevator); For students with FERPA waiver, graded homework can be picked up in the pick-up boxes in Downs (students without FERPA waiver: make specific arrangements with the TA on where to pick up your graded homework)

Late work will not be graded

You can discuss homework problems with other students in the class, but each of you should individually write up your own set of solutions. Collaboration on the exams is not allowed. 


Assignments
 Date PostedAssignment Due Date 
 9-25 HW1 10-2
 10-2 HW2 10-9
 10-9 HW3 10-16
 10-16 HW4 10-23
 10-23 Midterm 10-30
 10-30 HW5 11-6
 11-6 HW6 11-13
 11-13 HW7 11-20
 11-27 Final  12-6






Midterm and Final Exam

The midterm will be posted on 10-23 and is due on 10-30 (same schedule as the HW). The final exam will be posted on Monday 11-27 and is due on Wednesday 12-6 at 3PM. Follow the guidelines below regarding HW and exams. 

Note about Midterm and Final: it is NOT required that you use a blue book for the exams: you can use one if you wish, or you can write up your solutions in your preferred form (handwritten,  LaTeX written...): if you don't use a booklet make sure to staple your pages together before handing them in (we decline all responsibility for loose sheets of paper getting lost)


Collaboration Table
 HomeworkExams
You may consult:  
Course textbook (including answers in the back)YESYES
Other booksYESNO
Solution manualsNONO
InternetYESNO
Your notes (taken in class)YESYES
Class notes of othersYESNO
Your hand copies of class notes of othersYESYES
Photocopies of class notes of othersYESNO
Electronic copies of class notes of othersYESNO
Course handoutsYESYES
Your returned homework / examsYESYES
Solutions to homework / exams (posted on webpage)YESYES
Homework / exams of previous yearsNONO
Solutions to homework / exams of previous yearsNONO
Emails from TAsYESNO
You may:

Discuss problems with othersYESNO
Look at communal materials while writing up solutionsYESNO
Look at individual written work of othersNONO
Post about problems onlineNONO
For computational aids, you may use:

CalculatorsYES*NO
ComputersYES*NO

* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work.  For example, "by Mathematica" is not an acceptable justification for deriving one equation from another.  Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them.