Math 109a  
Introduction
to
Geometry
and
Topology 

Fall 201415  

Instructor:
Subhojoy Gupta,
304
Kellog, subhojoy @ caltech.edu TA: Ross Elliot, relliot @ caltech.edu

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No office hours on Tuesday Nov 25 and Tuesday Dec 2. Will be available on Wed afternoon 12pm instead. 
No office hours on Thu Oct 23. Will be available on Wed afternoon 12pm instead. 
Office hours on Oct 16 would be from 4:15  5pm. 
The first two classes (Sep 29 and Oct 1) shall be substituted
by M. Trnkova. 

Math 109a is the first course in the math 109 sequence,
Introduction to Geometry and Topology. In the first part of the course,
we shall introduce notions of general pointset topology, basic
examples and constructions.
Topics shall include the notions of compactness, metrizability,
separation properties, and completeness. The second part of the course
shall be an introduction to algebraic topology, including the
fundamental group and simplicial and singular homology. 

Prerequisites: Ma 2 or equivalent,
and Ma 108 must be taken previously or concurrently. Grading based on: homework, midterm and final exam Homework is due in the box each Friday at 10am . One late homework will be accepted, however the instructor needs to be informed before the due date. The late homework must be turned in with the next problem set. Collaboration is allowed for the homework but each should submit their own written work. No collaboration is allowed for the exams. 

Text: We will use "Topology" , James Munkres, ISBN: 0131816292. as a reference for the first part of the course. We will use Hatcher's "Algebraic Topology" (available at the author's webpage) as the textbook for the material in the second part of the course. 

This is a rough correspondence with topics in the textbooks (we sometimes did things a bit differently in class). The topics in italics were not covered in class, but are recommended reading for this course. Munkres: Chapter 2, with section 22 only till pg. 140. Chapter 3. except Theorem 26.9 and Example 3 of pg. 181. Chapter 4: except "Lindelof", "completely regular", "Sorgenfrey plane", Example 2 on pg. 203, "imbedding of manifolds". Chapter 7: except section 46, 47. Chapter 8: till pg. 298.  Hatcher: Chapter 0 : Except "join", "smash product", "reduced suspension", Proposition 0.19, Corollaries 0.20 and 0.21. Mapping cone, Proposition 0.18 Chapter 1, Section 1.1: All except: Theorem 1.8 (Fundamental Thm of Algebra), Corollary 1.11, Proposition 1.18. In addition we did: Equivalent of Section 59 (pg. 368) of Munkres, and a proof of the Jordan Separation Theorem. Chapter 1, Section 1.2: Proof of Van Kampen (we talked about the idea of the proof, and some special cases like the wedge product). Example 1.23, Corollary 1.28 and Example 1.29 Can omit (for now!): Examples 1.24, 1.25 Chapter 1, Section 1.3: We just saw the definition of covering spaces, some examples, and Propositions 1.30 and 1.31. We also saw how covering spaces arise by quotients under group actions. Page 58 is a good page to stare at. Chapter 2: Simplicial homology and computations, definition of singular homology groups, some properties and its homotopy invariance. We discussed idea of the proof of Theorem 2.10 (including the "prism operator"). Relative Homology groups: Pages 115119. We did not do the proof of Excision (Theorem 2.20) except the idea. We also did not do Theorem 2.13 (we computed the homology groups of spheres as in Corollary 2.14, differently.) Also, the following algebraic results were used without proof: Theorem 2.16 and The FiveLemma (pg. 129) . Last class: Brouwer Fixed Point Theorem (Corollary 2.15) and Invariance of Domain (Theorem 2.26). Links: Dunce hat A rope trick 


