Ben's Math 290 Page

I made a short version of the notes consisting of things I decided were particularly important to memorize. For a more complete collection of 290 notes, see below.

Qual prep problems.

• week 0 questions and solutions (cup product)
• week 1 questions and solutions (cup, cap, cross products, Poincare duality)
• week 2 questions and solutions (universal coefficients theorems, Kunneth, etc)
• week 3 questions and solutions (degree of maps between oriented manifolds)
• week 4 questions and solutions (fundamental group, covering spaces, Hurewicz)
• week 5 quals from 2009

Here are solutions sketches to much of the homework from winter quarter.

I should mention that I don't guarantee my notes and answers to be entirely correct!! I'm sure there are plenty of typos and probably a few larger errors of misunderstanding. Please let me know if you find errors, so that I can correct them, or at least flag them. You've been warned.

Nate Eldrege TA'd this course a few years ago. He still has some good material up on his web page. As does Ben Wilson from his 08 summer qual prep. And Tim Banham has a list of topology facts.

Justin assigned a homework problem to find a minimal triangulation for the torus. If you can find it on your own - awesome! But if, like me, you don't have the patience, here's a triangulation. (It's still interesting to prove that this is minimal!) Actually, Justin showed me a prettier way to display this triangulation using the dual to a "honeycomb" shape of six hexagons surrounding a center hexagon. I'll be happy to show you if you're interested.

If anyone in this class wants the TeX sources for any/all of the materials I'm posting online, just let me know and I'll be happy to give that to you.

From Winter quarter:

I highly recommend Munkres's Elements of Algebraic Topology. While it doesn't cover homotopy theory, it does a very good job with singular homology and cohomology.

When I took math 290, I typed up some notes to help me study for the qual. These didn't include any proof and they only roughly follow Nitu's course. I won't guarantee them to be correct, but if I'm providing them here in case they'd be a helpful:
math 290 notes.

One more word of warning: since these notes were basically intended for myself, I didn't always stick to standard conventions. In particular, composition of functions is defined in a funny way - please see the very first definition.

Thoughts on 1st homework (due 1/29):

• When I did problem 8, I think I used slightly stronger assumptions than were given in the problem. Feel free to do the same, just make a note of it in your write up.
• In problem 10, the formula L(sigma) = b(L(boundary(sigma))) is given for any sigma from Delta_n to Delta_n. I think this formula only really works for sigma = identity. But that's actually enough! The idea is to barycentrically divide the identity map in a controlled environment, and then push that out into the wild using a singular simplex. See p. 39 of my notes for a rewording of the problem. Or see Munkres's algebraic topology book for a full explanation.

Homework deadlines won't be strictly enforced. You should also feel free to hand in partially completed assignments.

Last updated: May 14

Up