| Week | Date | Topics covered | References | Homework assigned | Homework due |
|---|---|---|---|---|---|
| 1 | 1/14 | Brief overview of the course. Homotopy and homotopy equivalences. |
Hatcher § 0.Homotopy and Homotopy Type For future reference: Primer on category theory. |
HW1 - 0114 due 1/23 | |
| 1/16 | Pointed spaces. Pointed homotopies. Notes on group objects. |
Hatcher § 0.Homotopy and Homotopy Type May § 2.4 |
HW1 - 0116 due 1/23 | ||
| 1/18 | Relative homotopies. Spheres, discs, cones. Smash product. |
Hatcher § 0.Operations on spaces Hatcher § 0.Two criteria for homotopy equivalence May § 8.1, 8.2 |
HW1 - 0118 due 1/23 | ||
| 2 | 1/21 | MLK Day - No lecture. No office hour. | |||
| 1/23 | Suspension. Homotopy groups. |
Hatcher § 0.Operations on spaces Hatcher § 4.1.Definitions and basic constructions May § 8.2, 9.1 |
HW2 - 0123 due 1/30 | HW 1: Lectures 1/14, 1/16, 1/18 Solutions |
|
| 1/25 | Homotopy functors. Higher homotopy groups are abelian. |
Hatcher § 4.1.Definitions and basic constructions May § 9.1, 9.2 |
HW2 - 0125 due 1/30 | ||
| 3 | 1/28 | Effect of coverings. | Hatcher § 4.1.Definitions and basic constructions May § 9.4 |
HW3 - 0128 due 2/6 | |
| 1/30 | Dependence on basepoints. | Hatcher § 4.1.Definitions and basic constructions May § 9.5 |
HW3 - 0130 due 2/6 | HW 2: Lectures 1/23, 1/25 Solutions |
|
| 2/1 | Weak homotopy equivalences. CW-complexes. |
Hatcher § 4.1.CW approximation Hatcher § Appendix.Topology of cell complexes May § 9.5, 10.1, 10.2 |
HW3 - 0201 due 2/6 | ||
| 4 | 2/4 | Compactly generated spaces. | Hatcher § Appendix May § 5.1, 5.2 Notes on CGWH spaces by Neil Strickland |
HW4 - 0204 due 2/13 | |
| 2/6 | Cofibrations. Mapping cylinder. |
Hatcher § 0.The Homotopy Extension Property Hatcher § 4.H May § 6.1, 6.2 |
HW4 - 0206 due 2/13 | HW 3: Lectures 1/28, 1/30, 2/1 Solutions |
|
| 2/8 | More on cofibrations. Well-pointed spaces. |
Hatcher § 0.The Homotopy Extension Property Hatcher § 4.H May § 6.1, 6.2 |
HW4 - 0208 due 2/13 | ||
| 5 | 2/11 | Replacing a map by a cofibration. More on well-pointed spaces. |
Hatcher § 4.H May § 6.3, 8.3 |
HW5 - 0211 due 2/20 | |
| 2/13 | Relative homotopy groups. Long exact sequence of a pair. |
Hatcher § 4.1.Definitions and basic constructions May § 9.1, 9.2 |
HW5 - 0213 due 2/20 | HW 4: Lectures 2/4, 2/6, 2/8 Solutions |
|
| 2/15 | More on the LES and homotopy fibers. Fiber bundles. Intro to fibrations. |
Hatcher § 4.2.Fiber bundles May § 7.1 |
HW5 - 0215 due 2/20 | ||
| 6 | 2/18 | Fibrations. Change of fiber. |
Hatcher § 4.2.Fiber bundles May § 7.1, 7.2, 7.4 |
HW6 - 0218 due 2/27 | |
| 2/20 | Replacing a map by a fibration. Long exact sequence of a fibration. |
Hatcher § 4.2.Fiber bundles Hatcher § 4.3.Fibrations May § 7.3, 9.3 |
HW6 - 0220 due 2/27 | HW 5: Lectures 2/11, 2/13, 2/15 Solutions |
|
| 2/22 | (Towards the) Whitehead theorem. Compression lemma. n-connected maps. |
Hatcher § 4.1.Whitehead's Theorem May § 9.6 |
HW6 - 0222 due 2/27 | ||
| 7 | 2/25 | HELP: Homotopy extension and lifting property. Proof of the Whitehead theorem. |
Hatcher § 4.1.Whitehead's Theorem May § 9.6, 10.3 |
HW7 - 0225 due 3/6 | |
| 2/27 | Cellular approximation theorem. | Hatcher § 4.1.Cellular approximation May § 10.4 |
HW7 - 0227 due 3/6 | HW 6: Lectures 2/18, 2/20, 2/22 Solutions |
|
| 3/1 | CW approximation. | Hatcher § 4.1.CW Approximation May § 10.5 |
HW7 - 0301 due 3/6 | ||
| 8 | 3/4 | More on CW approximation. Fiber sequences. |
Hatcher § 4.3.Fibrations May § 8.6 |
HW8 - 0304 due 3/13 | |
| 3/6 | More on fiber sequences. Cofiber sequences. |
Hatcher § 4.3.The homotopy construction of cohomology May § 8.4 |
HW8 - 0306 due 3/13 | HW 7: Lectures 2/25, 2/27, 3/1 Solutions |
|
| 3/8 | Homotopy pullbacks. Old lecture notes / Expanded version. |
Hatcher § 4.H May § 10.7 |
HW8 - 0308 due 3/13 | ||
| 9 | 3/11 | Relation between fiber and cofiber sequences. Homotopy pushouts. Excisive triads. |
Hatcher § 4.H, 4.K May § 10.7 |
HW9 - 0311 due 3/27 | |
| 3/13 | Cartesian squares. Blakers-Massey homotopy excision theorem. Freudenthal suspension theorem. |
Hatcher § 4.2.Excision for Homotopy Groups May § 11.1, 11.2 |
HW9 - 0313 due 3/27 | HW 8: Lectures 3/4, 3/6, 3/8 Solutions |
|
| 3/15 | Stable homotopy groups. Proof of homotopy excision. |
Hatcher § 4.2.Excision for Homotopy Groups May § 11.3 |
HW9 - 0315 due 3/27 | ||
| 3/18 - 3/22 | Spring break - No lectures. | ||||
| 10 | 3/25 | LECTURE CANCELED. Hurewicz theorem. Homology Whitehead theorem. |
Hatcher § 4.2.The Hurewicz Theorem May § 15.1 The dual Whitehead theorems |
HW10 - 0325 due 4/3 | |
| 3/27 | Postnikov towers. Eilenberg-MacLane spaces. |
Hatcher § 1.B Hatcher § 4.1.CW Approximation Hatcher § 4.2.Excision for Homotopy Groups May § 15.Problems, 22.4 |
HW10 - 0327 due 4/3 | HW 9: Lectures 3/11, 3/13, 3/15 Solutions |
|
| 3/29 | Uniqueness of Eilenberg-MacLane spaces. Relation to cohomology. |
Hatcher § 4.2.Excision for Homotopy Groups Hatcher § 4.3.The Homotopy Construction of Cohomology May § 22.2 |
HW10 - 0329 due 4/3 | ||
| 11 | 4/1 | Hopf-Whitney theorem. k-invariants. |
Hatcher § 4.3.Postnikov Towers May § 22.4 |
HW11 - 0401 due 4/10 | |
| 4/3 | Obstruction theory via the Postnikov tower. | Hatcher § 4.3.Obstruction Theory | HW11 - 0403 due 4/10 | HW 10: Lectures 3/25, 3/27, 3/29 Solutions |
|
| 4/5 | Obstruction theory via the skeletal filtration. | May § 18.5 | HW11 - 0405 due 4/10 | ||
| 12 | 4/8 | Classifying spaces and universal bundles. | Hatcher § 4.2.Fiber Bundles May § 16.5, 23.1 |
HW12 - 0408 due 4/17 | |
| 4/10 | Construction of classifying spaces. | May § 16.5 Classifying spaces and infinite symmetric products Construction of universal bundles I and II |
HW12 - 0410 due 4/17 | HW 11: Lectures 4/1, 4/3, 4/5 Solutions |
|
| 4/12 | (Finish) Construction of classifying spaces. Symmetric products. |
Hatcher § 4.K | HW12 - 0412 due 4/17 | ||
| 13 | 4/15 | Dold-Thom theorem. | Hatcher § 4.K The Dold-Thom theorem |
HW13 - 0415 due 4/24 | |
| 4/17 | Brown representability theorem. Spectra. |
Hatcher § 4.E May § 22.2 Switzer § 9 up to Theorem 9.13 Cohomology theories |
HW13 - 0417 due 4/24 | HW 12: Lectures 4/8, 4/10, 4/12 Solutions |
|
| 4/19 | Topological K-theory. | Hatcher's VBKT May § 24.1 |
HW13 - 0419 due 4/24 | ||
| 14 | 4/22 | Bott periodicity. Hopf invariant. |
Hatcher's VBKT Hatcher § 4.B May § 24.2, 24.6 |
HW14 - 0422 due 5/3 | |
| 4/24 | More on the Hopf invariant. Homology of the James construction. |
Hatcher § 3.C | HW14 - 0424 due 5/3 | HW 13: Lectures 4/15, 4/17, 4/19 Solutions |
|
| 4/26 | Bott-Samelson theorem. | Hatcher § 4.J | No homework assigned today. | ||
| 15 | 4/29 | Intro to spectral sequences. | Hatcher's SSAT § 1.1 Introduction to spectral sequences |
HW14 - 0429 due 5/6 | |
| 5/1 | Last regular lecture. Serre spectral sequence. |
Hatcher's SSAT § 1.1, 1.2 | No homework assigned today. | Homework is due on Monday. | |
| 5/1 6 PM Room 243 AH |
Optional lecture. Serre classes of abelian groups. Mod C Hurewicz theorem. |
Hatcher's SSAT § 1.1.Serre Classes Mosher-Tangora § 10 |
|||
| 5/3 | No lecture. | Homework is due on Monday. | |||
| 16 | 5/6 | No lecture. | HW 14: Lectures 4/22, 4/24, 4/29 Solutions |
||
| Cutoff | Letter grade |
|---|---|
| 93.3 | A+ |
| 86.7 | A |
| 80 | A- |
| 73.3 | B+ |
| 66.7 | B |
| 60 | B- |
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