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 \(\alpha\).
- \(0 \to \mathbb{Z}/2 \to G \to \mathbb{Z} \to 0\)
- \(0 \to \mathbb{Z} \to G \to \mathbb{Z}/2 \to 0\)
- \(0 \to \mathbb{Z} \xrightarrow{\alpha} \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}/2 \to 0\)
- \(0 \to G \xrightarrow{\alpha} \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}/2 \to 0\)
- \(0 \to \mathbb{Z}/3 \to G \to \mathbb{Z}/2 \to \mathbb{Z} \xrightarrow{\alpha} \mathbb{Z} \to 0\)
- Hatcher p131 Section 2.1 Q4, 5, 14.
- 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 \(\alpha\).
- HW2, Due Feb 28th:
- Hatcher p131 Section 2.1 Q8, 17, 20, 29.
- HW3, Due Mar 16th:
- Compute the local homology groups \(H_*(X, X \setminus x)\), 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.
- The union of the unit sphere in \(\mathbb{R}^3\) with the parts of the \(x\)- and \(y\)-axes contained in the unit ball.
- The union of two round spheres in \(\mathbb{R}^3\) which intersect in a single circle.
- The union of the closed unit ball in \(\mathbb{R}^3\) with the closed ball of radius 2 in the \(x\)y-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 \(f^* \colon H^2(S^1 \times S^1; \mathbb{Z}) \to H^2(S^2; \mathbb{Z})\) is trivial for any map \(f \colon S^2 \to S^1 \times S^1\).
- Prove that \(f^* \colon H^2(S^1 \times S^1; \mathbb{Z}) \to H^2(K; \mathbb{Z})\) is trivial for any map \(f \colon K \to S^1 \times S^1\), 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:
These are the notes I make for class, they are probably not of much use to anyone else.