Math 189 -- Special Topics
Algebraic Topology
Spring 2012

Professor Francis Su
x73616, e-mail: mylastname at math.hmc.edu
Office Hours: THU 3:30-4:30 pm.

Course Content: This course is an introduction to algebraic and combinatorial topology, with an emphasis on simplicial and singular homology theory. A major theme in the course will be the connection between combinatorial and topological concepts. Topics will include simplicial complexes, simplicial and singular homology groups, exact sequences, chain maps, diagram chasing, Mayer-Vietoris sequences, Eilenberg-Steenrod axioms, Jordan curve theorem, and additional topics as time permits. This is standard first-year graduate material in pure mathematics.

Text: Munkres, Elements of Algebraic Topology. Supplemental reading: my book with Mike Starbird, Topology through Inquiry, available later. Doing the reading will be essential for success in this course.

Prerequisites: Analysis I (Math 131), and either Algebra I (Math 171) or Topology (Math 147), or permission of the instructor. Math 171 and 147 are recommended as co-requisites. This course is not a replacement for Math 147; it covers different material.

A note about the course: Expect this course to be challenging, but also quite rewarding, as you see the interplay between algebra, topology, combinatorics, and analysis. I will run this course more like a graduate course. As such, I will expect a certain level of mathematical maturity. This means that sometimes I will not prove simple statements in class; you may have to work out some details for yourself or by doing the reading. My focus will be on proving the larger theorems and providing perspective on the material.

Homeworks: Homeworks, assigned weekly, turn in bin outside my office by Fridays at 11am. Homeworks will be announced on the course webpage: http://www.math.hmc.edu/~su/math189/.

Grader: Dhruv Ranganathan

Honor Code: All are expected to abide by the HMC honor code. Cooperation is ENCOURAGED in this class, but write up all solutions individually and be sure to credit any collaborators.


HW Assignments
HW #1 (due 1/27) "R" means read and think about the problem. Only turn in problems NOT marked with an "R".

Problem A. Prove the statement in the text at the top of page 3: Points {a_0,...,a_n} are geometrically(affinely) independent iff the vectors a_1-a_0, a_2-a_0, ..., a_n-a_0 are linearly independent.

and Problems in Sections: 1 ( R1, R2, 3, R4, 5 ), 2 ( 1, 2, R3, 5, R6 ), 3 ( 1, 2 )

Talk to each other!! Call one person up that you do not know.

Notes: in 2.2, "path connected" means any 2 points are connected by a path.
In 3.1, B^2 is the 2-dim'l ball, and S^1 is the 1-dim'l sphere, a circle. To "identify" 2 points means to "make the same" or paste the two points together so they are the same point. To show the spaces are homeomorphic, please give a homeomorphism, and justify why it is a bi-continuous bijection (but I'm not expecting rigorous epsilon-delta arguments).
In 3.1, 3.2, "describe" means: sketch the underlying space (if possible) and describe its topological type (if you can, e.g., is it homeomorphic to some familiar space)?

If you haven't taken Math 147, feel free to look at these notes: notes from my Topology course (Math 147). I recommend reading Sections 4, 5, 6.
HW #2 (due 2/3) DO Section 2 ( 3 ) 3 ( 4 ) 4 ( R1, 2, 3ab, 3cd ) and 5 ( 1 ) and

Problem B. Prove Lemma 5.3 (use the def'n of a bdry map; look at the book's proof only if you need a hint).

Notes: in 2.3, "locally compact" means has a compact neighborhood (there's an open neighborhood whose closure is compact).

HW (due 2/10) Section 5 ( 2, 3, 4, 5, R6 ) and 6 ( 2, R4 )

HW (due 2/17) Section 6 ( 4, 6, 7, 8 ) 7 ( R1 ) 8 ( 1 )
HW (due 2/24) Section 9 ( 2, 3, G1, G5 ) 12 ( 1, 2 ). Here, 'G' means you don't have to give full justifications... just "guess" and give a sketch of your arguments using your intuition. (I'm trying to build your intuition.)
HW (due 3/2) Section 12 ( 3, 4 ) 13 ( 2[parts 1 to 4, read part 5], R4 )
before/after BREAK WEEK No Homework. But Skim Chapter 2.
HW (due 3/30) Section 15 ( R2 ) 19 ( 2, R3 ) 21 ( 1a, R1bc, R2 ) 23 ( 1 [statements (4) and (5) only], 3, 4 )

Please talk to each other about these problems!

HW (due 4/6) 23 ( 5 ) 24 ( 1, R2, 3a, 3b, 4, R5)

Please talk to each other about these problems!

HW (due 4/13) Section 25 ( 1, R4, R5 ) and
Problem C. Use the Mayer-Vietoris sequence to compute the homology of the connected sum of $n$ projective planes. (See Lecture 21 for a description of the space.)

Please talk to each other about these problems!

HW (due 4/20) Section 25 ( 4, 5 ) 26 ( 1a, R1b, R2 )
Problems BELOW this line MAY be assigned in the future, but assignments are only fixed if they are ABOVE this line.
HW (due ___) 29 ( 1a, R1b ) Section 29 ( 1 ) 30 ( 2, 4 ) 28 ( R2, R3 )
AND Do Problem D. Show that all universal objects in a category are equivalent.
HW (due ___) Section 33 ( 1 ) 21 ( 1, 2 )

In 33.1, I recommend breaking the space into two pieces X and Y, where X is the the topologist sine curve and Y is the line segment that connects the line part of X with the sine part of X. Please do this problem in two steps. Part (a): compute the homology of X. Part (b): use a Mayer-Vietoris sequence to compute the homology of T.