18.906 | Spring 2020 | Graduate

Algebraic Topology II


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.

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. Topology. Prentice Hall, 2000. ISBN: 9780139254956.

Neil Strickland’s notes on The Category of CGWH Spaces (PDF) (but in 2.12 he means the category CG, not CGWH).

Martin Frankland’s notes on Homotopy Theory (PDF).

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. A Short Course in Differential Topology. Cambridge University Press, 2018. ISBN: 9781108425797.

5 Fibrations, Fundamental Groupoid  tom Dieck, Tammo. Algebraic Topology. Corrected 2nd Printing, 2010. European Mathematical Society Publishing House, 2008. ISBN: 9783037190487.
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. Topology and Geometry (Graduate Texts in Mathematics 139). Springer, 1997. ISBN: 9780387979267.

 Varadarajan, Kalathoor.  The Finiteness Obstruction of C. T. C. Wall. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley and Sons, 1989. ISBN: 9780471623069.

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 Algebraic Topology (PDF - 3.8MB)
Vector Bundles and Principal Bundles  
16 Vector Bundles Husemöller, Dale. Fibre Bundles, 3rd edition. Springer, 1993. ISBN: 9780387940878.
17 Principal Bundles, Associated Bundles  <No suggested references>
18 I-invariance of BunG, and G-CW Complexes

Stephen A. Mitchell’s notes on Principal Bundles and Classifying Spaces.

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.” Mathematische Annalen. 262, no. 4 (1983) 487–501.

19 The Classifying Space of a Group  Knapp, Anthony William. Lie Groups Beyond an Introduction. 2nd ed. Birkhäuser, 2002. ISBN:  9780817642594.
20 Simplicial Sets and Classifying Spaces

Milnor, John. “The Geometric Realization of a Semi-Simplicial Complex (PDF).” Annals of Mathematics 65, No. 2 (1957) 357–362.

Goerss, Paul and Jardine, John. “Simplicial Homotopy Theory (PDF - 3.9MB).” Progress in Mathematics 174, Birkhäuser Verlag, 1999.

21 The Čech Category and Classifying Maps Segal, Graeme. 1968. “Classifying Spaces and Spectral Sequences.” Publications Mathématiques. 27 (34): 105–112.
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 
<No suggested references>  
27 The Serre Exact Sequence and the Hurewicz Theorem Spanier, Edwin H. Algebraic Topology. Springer, 1966. ISBN: 9780387944265. (and later reprints)
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, Contemporary Mathematics 146 (1993) 1–30.

Bousfield, Aldridge Knight. “The Localization of Spaces with Respect to Homology (PDF - 1.1MB).” Topology 14 (1975) 133–150.

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. Notes on Cobordism Theory. Princeton University Press, 2015. ISBN: 9780691622217. (Originally published in 1968)

Thom, René. “Quelques Propriétés Globales des Variétés Différentiables (PDF).” Commentarii Mathematici Helvitici 28 (1954) 17–86.

Atiyah, Michael. “Thom Complexes.” Proceedings of the London Philosophical Society 11 (1961) 291–310.

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.” Proceedings of the Cambridge Philosophical Society 57 (1961) 200–208.

Milnor, John. “A Procedure for Killing Homotopy Groups of Differentiable Manifolds (PDF).” Proceedings of Symposia in Pure Mathematics, III (1961) 39–55.

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Spring 2020
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