The Berstein seminar (MATH 7520) this semester will be on the topic of principal G-bundles and classifying spaces.

Announcements

• The first class will take place on Tuesday, January 22.

Topics covered

• Week 1 (01/24, 01/26): Crash course on classical bundle theory: fiber bundles, vector bundles, principal G-bundles, characteristic classes.
• Week 2 (01/29, 01/31): Associated bundles and their sections, homotopy classification of principal bundles, passing from principal bundles to other types of bundles.
• Week 3 (02/05, 02/07): Universal principal bundles, B as a functor, k-connectivity of fibers and global sections, classical examples of classifying spaces.
• Week 4 (02/12, 02/14): Models for classifying spaces (colimits, smooth analytic models, simplicial), Milnor's construction of BG (naively).
• Week 5 (02/19, 02/21): Re-examining Milnor's construction of BG, what kinds of objects can be classified by "BG"-like constructions?
• Week 5 (02/26, 02/28): Properties of BG and calculational tools, the sheaf-theoretic perspective.
• Week 6 (03/05, 03/07): Arizona Winter School.
• Week 7 (03/12, 03/14): Algebraic vs. topological bundles, the Borel–Weil–Bott theorem and its applications.
• Week 8 (03/19, 03/21): The classifying space of a category, the categorical bar and cobar construction.
• Week 9 (03/26, 03/28): Bar and cobar constructions for dg algebras, Kan extensions, Dold–Kan correspondence.
• Week 10 (04/09, 04/11): Relation between the algebra and topology of classifying spaces and the algebra and geometry of moduli spaces.
• Week 11 (04/16, 04/18): Introduction to equivariant K-theory and its connection to principal G-bundles.
• Week 12 (04/23, 04/25): Localization theorems in equivariant K-theory and its relation to representation theory a la Beilinson–Bernstein and Kazhdan–Lusztig.
• Week 13 (04/30, 05/02): Spectral sequences from (co)simplicial objects, filtrations from skeleta, and applications.
• Week 14 (05/07, 05/09): The Atiyah–Segal completion theorem and its generalizations, G-structures and vistas.

Readings

Principal Bundles and Some Applications

• Borel and Hirzebruch, "Characteristic classes and homogeneous spaces. I," Amer. J. Math. 80 (1958), Ch. III.

Chern–Simons theory

• Chern and Simons, "Some cohomology classes in principal fiber bundles and their application to riemannian geometry." Proc. Nat. Acad. Sci. U.S.A. 68 (1971) 791–794.
• Chern and Simons, "Characteristic Forms and Geometric Invariants," Ann. of Math. (2) 99 (1974), 48–69.

Borel–Weil–Bott Theorem

• Demazure, "Une démonstration algébrique d'un théorème de Bott," Invent. Math. 5 (1968) 349–356.
• Demazure, "A very simple proof of Bott's theorem," Invent. Math. 33 (1976), no. 3, 271–272.

Equivariant K-theory

• Segal, "Equivariant K-theory," Publications Mathématiques de l'IHÉS, Volume 34 (1968), p. 129-151
• Atiyah and Segal, "Equivariant K-theory and completion," J. Differential Geometry 3 (1969) 1–18.

Obstruction Theory

• Hermann, "Obstruction theory for fibre spaces," Illinois J. Math. 4 (1960) 9–27.
• Hermann, "Secondary obstructions for fibre spaces," Bull. Amer. Math. Soc. 65 (1959) 5–8.
• Stewart, "Lifting group actions in fibre bundles," Ann. of Math. (2) 74 (1961) 192–198.
• Robinson, "Moore–Postnikov systems for non-simple fibrations," Illinois J. Math. 16 (1972), 234–242.

Classifying Spaces

Explicit realization

• Milnor, "Construction of universal bundles, I," Ann. of Math. (2) 63 (1956), 272–284.
• Milnor, "Construction of universal bundles. II," Ann. of Math. (2) 63 (1956), 430–436.

Bar construction

• Segal, "Classifying spaces and spectral sequences," Inst. Hautes Etudes Sci. Publ. Math. (1968), no. 34, 105–112
• May, "Classifying spaces and fibrations," Mem. Amer. Math. Soc. 1 (1975), no. 1, 155, Ch. 7 and 8.

H-Spaces and Infinite Loop Spaces

• Dold and Lashof, "Principal quasi-fibrations and fibre homotopy equivalence of bundles," Illinois J. Math. 3 (1959) 285–305.


• Stasheff, "Homotopy associativity of H-spaces, I and II," Trans. Amer. Math. Soc. 108 (1963), 275-292; 293–312.


• Boardman and Vogt, "Homotopy-everything H-spaces," Bull. Amer. Math. Soc. 74 (1968) 1117–1122.


• Barratt and Priddy, "On the homology of non-connected monoids and their associated groups," Comment. Math. Helv. 47 (1972), 1–14.
• Quillen, "On the group completion of a simplicial monoid," preprint.


• McDuff and Segal, "Homology fibrations and the 'group-completion' theorem," Invent. Math. 31 (1975/76), no. 3, 279–284.

Back to Brian Hwang's home page.