Original figures from Poincare's 5th l’Analysis Situs paper, where the Poincare conjecture is posed. The last figure depicts the Poincare homology sphere.

This is a graduate level introduction to algebraic topology with a focus on homology and its applications. This course is intended to prepare PhD students for the screening exam and research.

Topics will include the fundamental group, singular simplicial and cellular homology, Eilenberg-Steenrod axioms, homological algebra, homotopy groups, covering spaces and characteristic classes.

We will assume a background in undergraduate real analysis, point-set topology, commutative algbera and linear algebra.

• Instructor: Julian Chaidez (julian.chaidez@usc.edu)
• Assistant: Tina Peng (pengtina@usc.edu)
• Textbooks: An Introduction To Algebraic Topology by Rotman and Characteristic Classes by Milnor-Stasheff.
• Lectures: MWF 11-12a in KAP 414
• Office Hours: MF 4-5 in KAP 245
• Grade Scheme: 40% HW + 20% midterm + 40% final.
• Exam Structure: Each exam will consist of a take home and in class component, weighted equally.
• In-Class Exam Dates: 10/14 (Midterm) and 12/11 (Final)
• For more information, see the course syllabus.

• HW1 (9/4). 0.3, 0.4, 0.6, 0.12, 0.15, 0.18, 1.1, 1.2, 1.5
• HW2 (9/11). 2.4, 3.8, 3.19, 3.21, 3.22, 4.2, 4.3
• HW3 (9/18). 4.4, 4.6, 4.10, 4.11, 4.13, 5.1, 5.2, 5.3, 5.4, 5.7, 5.8
• HW4 (9/25). 5.11, 5.13, 5.15, 5.20, 5.21, 6.4, 6.5, 6.21, 6.23
• HW5 (10/2). 7.2, 7.3, 7.10, 7.19, 7.20, 7.26, 8.2, 8.17, 8.36
• Midterm (10/14). Take home component.
• HW6 (10/23). 9.48, 4.13, 4.14, 7.47, 10.1, 10.3, 10.4
• HW7 (10/30). 10.15, 10.16, 10.18, 10.19, 10.33, 11.8, 11.9
• HW8 (11/6). 11.12, 11.24, 11.25, 11.29, 11.31, 11.32
• HW9 (11/13). 12.1, 12.2, 12.3, 12.15, 12.17, 12.18, 12.19
• HW10 (11/25). MS A-2, 1-A, 1-B, 2-A, 2-B, 3-B, 3-E, 4-B, 4-C
• Final (12/6). Take home component.

• M 8/26. overview, fixed point theorem, categories and functor. 1-13
• W 8/28. homotopy, contractibility, cones, connectedness. 14-30
• F 8/30. simplicies, affine spaces, affine maps. 31-39
• W 9/4. fundamental groupoid, the fundamental group functor. 39-56
• F 9/6. singular complex, singular homology functor. 57-68.
• M 9/9. dimension axiom, homotopy axiom, Hurwicz (statement). 68-80
• W 9/11. complexes, chain maps, chain homotopies. 86-93.
• F 9/13. exact sequences, relative homology. 93-105
• M 9/16. excision, Mayer-Vietoris, start excision proof. 105-119
• W 9/18. finish excision, applications of excision. 111-130
• F 9/20. simplicial complexes, simplicial approximation. 131-142
• M 9/23. simplicial homology,comparison with singular. 142-164
• W 9/25. cell attachments, CW complexes. 180-212
• F 9/27. cellular homology. 212-227
• M 9/30. natural transformations, Eilenberg-Steenrod axioms. 228-237
• W 10/2. tensor products, universal coefficients. 253-265
• F 10/4. Eilenberg-Zilber theorem, Kunneth formula. 265-272
• M 10/7. Hurewicz map, Hurewicz theorem. 80-85
• W 10/9. Van Kampen, computations. 164-180
• W 10/16. covering spaces, basic properties. 272-284.
• F 10/18. covering transformations. 284-295
• M 10/21. existence, orbit. 295-311
• W 10/23. function spaces, (co)group objects. 312-323
• F 10/25. loop space, suspension. 323-334
• M 10/28. higher homotopy groups. 334-344
• W 10/30. exact sequence of a pair & 344-355
• F 11/1. exact sequence of a fibration. 355-368
• W 11/6. cohomological algebra, singular cohomology. 377-390
• F 11/8. graded algebras, cup product, cap product. 390-402
• W 11/13. topological manifolds, fundamental class. MS 270-275
• F 11/15. compactly supported cohomology, Poincare duality. MS 275-280
• M 11/18. smooth manifolds, examples, constructions. MS 3-13
• W 11/20. vector bundles, examples, constructions. 12-37
• F 11/22. Stieffel-Whitney classes, applications. 37-55.
• M 11/25. oriented bundles, Euler class. MS 95-105
• M 12/2. the Thom isomorphism theorem. MS 105-115
• W 12/4. complex vector bundles, complex manifolds. MS 149-155
• F 12/6. Chern classes, properties, computations. MS 155-173