This section provides references to cover details not discussed in lecture. These are not required readings, but may be helpful in deepening your understanding of the subject.

LEC # | TOPICS | REFERENCES |
---|---|---|

Basic Homotopy Theory | ||

1 | Limits, Colimits, and Adjunctions | Mac Lane, Saunders. Categories for the Working Mathematician. 2nd ed. Springer, 2010. ISBN: 9780387984032. |

2 | Cartesian Closure and Compactly Generated Spaces | Hatcher, Allen. Algebraic Topology. Cambridge University Press, 2009. ISBN: 9780521795401. Munkres, James Raymond. Neil Strickland's notes on Martin Frankland’s notes on |

3 | Basepoints and the Homotopy Category | Fritsch, Rudolf, and Renzo A. Piccinini. Cellular Structures in Topology. Cambridge University Press, 1990. ISBN: 9780521327848. |

4 | Fiber Bundles | An animation of fibers in the Hopf fibration over various points on the two-sphere by Niles Johnson. "Hopf fibration -- fibers and base." YouTube. Dundas, Bjørn Ian. |

5 | Fibrations, Fundamental Groupoid | tom Dieck, Tammo. |

6 | Cofibrations | <No suggested references> |

7 | Cofibration Sequences and Co-exactness | <No suggested references> |

8 | Weak Equivalences and Whitehead’s Theorems | <No suggested references> |

9 | Homotopy Long Exact Sequence and Homotopy Fibers | Strøm, Arne. "A Note on Cofibrations (PDF)." Mathematica Scandinavica 19 (1966) 11–14. |

The Homotopy Theory of CW Complexes | ||

10 | Serre Fibrations and Relative Lifting | Stephen A. Mitchell's notes on Serre Fibrations (PDF). |

11 | Connectivity and Approximation | Bredon, Glen. Varadarajan, Kalathoor. |

12 | Cellular Approximation, Obstruction Theory | <No suggested references> |

13 | Hurewicz, Moore, Eilenberg, Mac Lane, and Whitehead | <No suggested references> |

14 | Representability of Cohomology | Brown, Edgar. "Cohomology Theories (PDF - 1.3MB)." Annals of Mathematics 75 (1962) 467–484. |

15 | Obstruction Theory | James Davis' and Paul Kirk's notes on |

Vector Bundles and Principal Bundles | ||

16 | Vector Bundles | Husemöller, Dale. |

17 | Principal Bundles, Associated Bundles | <No suggested references> |

18 | I-invariance of Bun_{G}, and G-CW Complexes | Stephen A. Mitchell's notes on Lück, Wolfgang. "Survey on Classifying Spaces for Families of Subgroups." (2005) 269–322. Illman, Sören. "The Equivariant Triangulation Theorem for Actions of Compact Lie Groups." |

19 | The Classifying Space of a Group | Knapp, Anthony William. |

20 | Simplicial Sets and Classifying Spaces | Milnor, John. "The Geometric Realization of a Semi-Simplicial Complex (PDF)." Goerss, Paul and Jardine, John. "Simplicial Homotopy Theory (PDF - 3.9MB)." |

21 | The Čech Category and Classifying Maps | Segal, Graeme. 1968. "Classifying Spaces and Spectral Sequences." |

Spectral Sequences and Serre Classes | ||

22 | Why Spectral Sequences? | Miller, Haynes. "Leray in Oflag XVIIA: The Origins of Sheaf Theory, Sheaf Cohomology, and Spectral Sequences (PDF)." Gazette des Mathematiciens 84 suppl (2000) 17–34. |

23 | The Spectral Sequence of a Filtered Complex | <No suggested references> |

24 | Serre Spectral Sequence | Serre, Jean-Pierre. "Homologie Singulière des Espaces Fibrés (PDF - 6.2MB)." Applications. Annals of Mathematics 54 (1951), 425–505. |

25 | Exact Couples | <No suggested references> |

26 | The Gysin Sequence, Edge Homomorphisms, and the Transgression | <No suggested references> |

27 | The Serre Exact Sequence and the Hurewicz Theorem | Spanier, Edwin H. |

28 | Double Complexes and the Dress Spectral Sequence | Dress, A. "Zur Spectralsequenz von Faserungen." Inventiones mathematicae 3 (1967): 172-178. |

29 | Cohomological Spectral Sequences | <No suggested references> |

30 | Serre Classes | <No suggested references> |

31 | Mod C Hurewicz and Whitehead Theorems | <No suggested references> |

32 | Freudenthal, James, and Bousfield | Miller, Haynes and Douglas Ravenel. "Mark Mahowald’s Work on the Homotopy Groups of Spheres (PDF)." Algebraic Topology, Oaxtepec 1991, Bousfield, Aldridge Knight. "The Localization of Spaces with Respect to Homology (PDF - 1.1MB)." |

Characteristic Classes, Steenrod Operations, and Cobordism | ||

33 | Chern Classes, Stiefel-Whitney Classes, and the Leray-Hirsch Theorem | <No suggested references> |

34 | H*(BU(n)) and the Splitting Principle | <No suggested references> |

35 | The Thom Class and Whitney Sum Formula | <No suggested references> |

36 | Closing the Chern Circle, and Pontryagin Classes | <No suggested references> |

37 | Steenrod Operations | <No suggested references> |

38 | Cobordism | Stong, Robert Evert. Thom, René. "Quelques Propriétés Globales des Variétés Différentiables (PDF)." Atiyah, Michael. "Thom Complexes." |

39 | Hopf Algebras | Milnor, John. "The Steenrod Algebra and its Dual (PDF - 1.4MB)." Annals of Mathematics 67 (1958) 150–171. |

40 | Applications of Cobordism | Atiyah, Michael. "Bordism and Cobordism." Milnor, John. "A Procedure for Killing Homotopy Groups of Differentiable Manifolds (PDF)." |