MA 532 Algebraic Topology
Email: spercival@unm.edu
Syllabus: Syllabus
Office Location: SMLC 220
Wednesday: 11:00-12:00
Schedule
| Date | Topic(s) | Reference(s) |
|---|---|---|
| Monday, August 18, 2025 | Groups | DF 1.1-1.5 |
| Wednesday, August 20, 2025 | Homomorphisms, group actions, subgroups | DF 1.6, 1.7, 2.1 |
| Friday, August 22, 2025 | Subgroups | DF 2.1, 2.2 |
| Monday, August 25, 2025 | Cyclic groups, subgroups generated by a subset | DF 2.3, 2.4 |
| Wednesday, August 27, 2025 | Lattice of subgroups, cosets and quotient groups | DF 2.5, 3.1 |
| Friday, August 29, 2025 | Quotient groups, Lagrange's Theorem | DF 3.1, 3.2 |
| Monday, September 1, 2025 | Labor day, no class | |
| Wednesday, September 3, 2025 | Isomorphism Theorems, composition series | DF 3.3, 3.4 |
| Friday, September 5, 2025 | Composition series, the symmetric group | DF 3.4, 1.3, Gallian chapter 5 |
| Monday, September 8, 2025 | The alternating group, group actions and permutation representations | Gallian chapter 5, DF 4.1 |
| Wednesday, September 10, 2025 | Cayley's Theorem | DF 4.2 |
| Friday, September 12, 2025 | The class equation | DF 4.3 |
| Monday, September 15, 2025 | Automorphisms | DF 4.4 |
| Wednesday, September 17, 2025 | Sylow's Theorem | DF 4.5 |
| Friday, September 19, 2025 | Sylow's Theorem | DF 4.5 |
| Monday, September 22, 2024 | The Simplicity of A_n | DF 4.6 |
| Wednesday, September 24, 2024 | Review | |
| Friday, September 26, 2024 | Exam 1 | |
| Monday, September 29, 2025 | Direct Products | DF 5.1 |
| Wednesday, October 1, 2025 | Free groups, the Fundamental Theorem of Finitely Generated Abelian Groups | DF 6.3, 5.2 |
| Friday, October 3, 2025 | The Fundamental Theorem of Finitely Generated Abelian Groups, Recognizing direct products | DF 5.2, 5.4 |
| Monday, October 6, 2025 | Semidirect products | DF 5.5 |
| Wednesday, October 8, 2025 | Semidirect products | DF 5.5 |
| Friday, October 10, 2025 | Fall break; no class | |
| Monday, October 13, 2025 | p-groups, nilpotent groups, and solvable groups | DF 6.1 |
| Wednesday, October 15, 2025 | p-groups, nilpotent groups, and solvable groups | DF 6.1 |
| Friday, October 17, 2025 | p-groups, nilpotent groups, and solvable groups | DF 6.1 |
| Monday, October 20, 2025 | Introduction to rings | DF 7.1 |
| Wednesday, October 22, 2025 | Polynomial rings, matrix rings, and group rings | DF 7.2 |
| Friday, October 24, 2025 | Ring homomorphisms and quotient rings | DF 7.3 |
| Monday, October 27, 2025 | Ring homomorphisms and quotient rings | DF 7.3 |
| Wednesday, October 29, 2025 | Review | |
| Friday, October 31, 2025 | Exam 2 | |
| Monday, November 3, 2025 | Properties of ideals | DF 7.4 |
| Wednesday, November 5, 2025 | Properties of ideals, rings of fractions | DF 7.4, 7.5 |
| Friday, November 7, 2025 | Rings of fractions | DF 7.5 |
| Monday, November 10, 2025 | Chinese Remainder Theorem | DF 7.6 |
| Wednesday, November 12, 2025 | Euclidean domains | DF 8.1, Z[ω] |
| Friday, November 14, 2025 | PIDs | DF 8.2 |
| Monday, November 17, 2025 | Unique factorization domains | DF 8.3 |
| Wednesday, November 19, 2025 | Polynomial rings (over fields), Polynomial rings that are UFDs | DF 9.1-9.3 |
| Friday, November 21, 2025 | Irreducibility criteria, Polynomial rings over fields II | DF 9.4, 9.5 |
| Monday, November 24, 2025 | Introduction to module theory | DF 10.1 |
| Wednesday, November 26, 2025 | No class | |
| Friday, November 28, 2025 | No class | |
| Monday, December 1, 2025 | Quotient modules and module homomorphisms | DF 10.2 |
| Wednesday, December 3, 2025 | Free modules | DF 10.3 |
| Friday, December 5, 2025 | Review |
Homework
*Starred problems are not graded but strongly recommended. Please do NOT turn these in with your homework.
Homework 1: Due Friday, August 29
DF 1.1.31, 1.2.6, 1.4.10, 1.4.11 (NOT part c), 1.5.3, 1.6.2, 1.6.11, 1.7.7, 2.1.4, 2.2.3
Homework 2: Due Friday, September 5
DF 2.3.15 (hint: you may use any of the previous problems in this section without proof), 2.3.16, 2.3.24, 2.4.14, 2.4.19, 2.5.9(b), 3.1.3, 3.1.22 (you may use the fact that the intersection of an arbitrary nonempty collection of subgroups of G is itself a subgroup of G), 3.1.27
Homework 3: Due Friday, September 12
DF 3.2.4*, 3.2.8, 3.2.10 (hint: you may use any other exercise in this section without proof), 3.2.18*, 3.3.2 (worth 8 points), 3.4.6, 3.4.11*
Homework 4: Due Friday, September 19
DF 3.5.4, 3.5.10, 3.5.13, 4.1.2 (hint: you may use any other exercise in this section without proof), 4.1.3, 4.2.11, 4.2.12*, 4.2.13 (hint: use 4.1.12)
Homework 5: Due Friday, October 3
DF 4.3.2(a), 4.3.13*, 4.3.19, 4.3.20*, 4.4.8, 4.5.6, 4.5.7*, 4.5.13*, 4.5.17, 4.5.26*, 4.5.30, 4.5.32*, 4.5.33, 4.5.37*, 4.5.40*
Homework 6: Due Friday, October 17
DF 5.1.1, 5.1.4, 5.1.14. 5.2.1b, 5.2.8*, 5.2.9, 5.4.16, 5.4.19*, 5.5.1, 5.5.5 (see p 179 for definition of Hol), 5.5.6, 5.5.7*, 5.5.11
Homework 7: Due Friday, October 24
DF 2.4.17*, 6.1.1, 6.1.6, 6.1.8, 6.1.9, 6.1.12, 6.1.14, 6.1.18*, 6.1.19*, 6.1.21, 6.1.22*
Homework 8: Due Friday, November 7
DF 7.1.11, 7.1.12*, 7.1.13, 7.1.14*, 7.1.15, 7.1.16*, 7.1.21, 7.1.22*, 7.2.2*, 7.3.4, 7.3.8*, 7.3.10, 7.3.15*, 7.3.26, 7.3.27*, 7.3.28*, 7.3.33 (you may assume problem 7.1.14 without proof)
Homework 9: Due Friday, November 14
DF 7.4.8, 7.4.10, 7.4.11*, 7.4.15, 7.4.19*, 7.4.23*, 7.4.24*, 7.4.25, 7.4.26*, 7.4.27*, 7.4.28*, 7.4.30*, 7.4.31*, 7.4.32*, 7.5.1*, 7.5.5*, 7.6.5(a), 7.6.7 (1 test point extra credit)
Homework 10: Due Friday, November 21
DF 8.1.3, 8.1.10, 8.1.11, 8.2.1, 8.2.2*, 8.2.5, 8.2.6, 8.2.7
Homework 11: Due Friday, December 5
DF 8.3.2, 8.3.6, 8.3.8, 8.3.11*, 9.1.6, 9.2.1, 9.2.2, 9.2.5, 9.2.6*, 9.2.7, 9.2.8*, 9.3.1, 9.3.2*, 9.3.3, Extra credit: If \((I, x) \subset R[x]\) is maximal, must \(I \subset R\) be maximal? 1 test point for a solution sketch, 1 additional test point for full solution.
Suggested module exercises: Do not turn in.
DF 10.1.4, 10.1.5, 10.1.6, 10.1.7, 10.1.8, 10.1.9, 10.1.10, 10.2.2, 10.2.5, 10.2.6, 10.2.9, 10.2.10, 10.3.2, 10.3.4, 10.3.7, 10.3.11, 10.3.12, 10.3.27
