Math 70800, Topology I, Spring 2023

TTh 11:45 - 1:15 Room 5417

Instructor: Joseph Maher
Office: 4308
Office hours: T 1:15-2:15, Th 1:15-2:15
Webpage: http://www.math.csi.cuny.edu/~maher/teaching
Email: joseph.maher@csi.cuny.edu
Phone: (718) 982-3623

Text: Allen Hatcher, Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0.

Outline:

We will carry on from where the Fall semester Topology I 70700 left off. We will aim to cover Chapters 2 and 3 from Hatcher, and more if we have time.

Topics:

  • Homology: simplicial/singular/cellular homology, exact sequences, Mayer-Vietoris, Brouwer fixed point theorem, Hurewicz.

  • Cohomology: universal coefficients, cup products, Kunneth formula, Poincare duality.

Homework:

We will have homework every other week.

  • HW1, Due Feb 9th:
    • For each of the following exact sequences of abelian groups and homomorphisms, say as much as you can about the unknown group G, and/or the unknown homomorphism α.
      1. 0 → ℤ/2 → G → ℤ → 0
      2. 0 → ℤ → G → ℤ/2 → 0
      3. $0 \to \mathbb{Z} \xrightarrow{\alpha} \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}/2 \to 0$
      4. $0 \to G \xrightarrow{\alpha} \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}/2 \to 0$
      5. $0 \to \mathbb{Z}/3 \to G \to \mathbb{Z}/2 \to \mathbb{Z} \xrightarrow{\alpha} \mathbb{Z} \to 0$
    • Hatcher p131 Section 2.1 Q4, 5, 14.
  • HW2, Due Feb 28th:
    • Hatcher p131 Section 2.1 Q8, 17, 20, 29.
  • HW3, Due Mar 16th:
    • Compute the local homology groups H*(X, X \ x), where x is the central vertex of the graph consisting of three edges meeting at a single vertex.
    • Describe explicit cell structures on the following spaces.
      1. The union of the unit sphere in ℝ3 with the parts of the x- and y-axes contained in the unit ball.
      2. The union of two round spheres in ℝ3 which intersect in a single circle.
      3. The union of the closed unit ball in ℝ3 with the closed ball of radius 2 in the xy-plane.
    • Hatcher p156 Section 2.2 Q9, 12.
  • HW4, Due Apr 4th:
    • Hatcher p156 Section 2.2 Q10, 28, 29, 30
  • HW5, Due Apr 27th:
    • Hatcher p205 Section 3.1 Q5, 6, 3.2 Q1, 3, 7
    • Prove that f*: H2(S1 × S1; ℤ) → H2(S2; ℤ) is trivial for any map f: S2 → S1 × S1.
    • Prove that f*: H2(S1 × S1; ℤ) → H2(K; ℤ) is trivial for any map f: K → S1 × S1, where K is the Klein bottle.
  • HW6, Due May 11th:
    • Hatcher 3.2 Q11, 3.3 Q7, 10, 11, 31, 32

Here are some of the old quals:

Quals

These are the notes I make for class, they are probably not of much use to anyone else.