Lecture Schedule / News and Updates
- Fri 5/4: Exams handed back. Wrap up.
The characteristic class notes are probably about as far along as they're going to get this semester.
- Wed 5/2: Proof of Leray-Hirsch theorem, Chern classes. Hatcher pp. 432-434.
- Mon 4/30: Computation of H^*(BO(n),Z/2) using the Leray Hirsch theorem. Hatcher pp. 435-437.
With regard to problem 4(b): it is probably false in general. I leave it to you to formulate some condition on G for which it does hold.
- Fri 4/27: Stiefel-Whitney classes and some computations.
The notes are a bit behind but are being updated slowly but surely.
The take-home final exam is ready and will be due on Wednesday 5/2.
- Wed 4/25: The setting for characteristic classes. B as a functor.
- Mon 4/23: Universal principal bundles.
- Fri 4/20: Homotopy properties and classification of principal bundles.
Here are notes for this week's lectures.
- Wed 4/18: Principal bundles.
- Mon 4/16: Intro. to characteristic classes: Fiber bundles, vector bundles
and principal bundles.
- Mon 4/9: Fibrations, fiber sequences and the LES in homotopy groups for a fibration. May Ch. 6, 7.5-7.6,. Hatcher pp. 405-409.
- Fri 4/6: Cohomology as a representable functor concluded.
There will be no class on Wednesday 4/11 or Friday 4/13 next week.
- Wed 4/4: Cohomology as a representable functor. Cofiber/Puppe sequences. Hatcher pp. 393-398. May Ch. 8.1-8.4.
Problem set 4 has been posted. It will be due Mon 4/9.
- Mon 4/2: Compactly generated spaces, cofibrations and fibrations. May: Ch 5, 6.1, 6.2.
- Mon 3/26-Fri 3/30: Spring break, no class.
- Fri 3/23: CW approximation continued. pp. 352-357.
Solutions to Problem Set 3 have been written.
- Wed 3/21: CW approximation. pp. 352-357.
- Mon 3/19: Cellular approximation. pp. 348-351.
- Fri 3/16: Long exact homotopy sequence of a pair, Whitehead's theorem. pp. 344-348.
- Wed 3/14: Higher homotopy groups. pp. 339-344.
- Mon 3/12: Applications of duality,
intro to homotopy theory.
- Fri 3/9: Poincare duality. pp. 246-248.
- Wed 3/7: Cohomology with compact supports. pp. 242-245.
Problem set 3 has been posted and will be due on Monday 3/12.
- Mon 3/5: Fundamental classes, continued. Introducing the cap product. pp. 236-241.
- Fri 3/3: Orientation covering spaces, fundamental classes for orientable manifolds. pp. 234-236.
- Wed 2/29: Wrap up on products in cohomology. Orientation and intro to Poincare duality. pp. 230-234.
The notes have been updated once again, to cover the Kunneth theorem in homology and the cross/cup products in cohomology.
- Mon 2/27: Kunneth theorem, graded commutativity of cross products, the cup product via the cross product.
- Fri 2/24:
Kunneth theorems.Eilenberg-Zilber theorem continued, the exterior product in homology.
The notes have been updated to cover today's lecture.
- Wed 2/22:
Graded commutativity of cup product,tensor products of chain complexes and the Eilenberg-Zilber theorem.
Here are some notes covering what I did in lecture today. They will be expanded to cover Friday's lecture also.
- Mon 2/20: No class.
- Fri 2/17: Cup product computations for projective spaces,
graded commutativity of cup product.
Erratum: I stated in class that a good pair (Y,B) was homotopy equivalent to the pair (Y/B, B/B). That is clearly false in general; rather what I should have said was that the quotient map induces an isomorphism on (co)homology groups using excision.
- Wed 2/15: Cup product computations for the projective plane and the torus, possibly RP^n.
- Mon 2/13: The cup product. pp. 206-210.
- Fri 2/10: Some applications of homology with coefficients, field coefficients and duality,
introduction to products in cohomology. pp. 153-155, 198, 266-267.
- Wed 2/8: Homology with coefficients, tensor products, Tor and the UCT for homology.
- Mon 2/6: Ext continued, Universal Coefficient Theorem (UCT) for cohomology.
The notes have been updated.
- Fri 2/3: Ext as a derived functor.
Here are some notes on Ext, essentially covering today's lecture. They will be expanded to cover Tor and the universal coefficient theorems next week.
- Wed 2/1: Definition of singular cohomology, derivation of the properties. pp. 197-202. (Hatcher proves the universal coefficient theorem before the properties of cohomology, while we will proceed in the opposite order.)
- Mon 1/30: Axioms for cohomology as a functor; reduced vs. unreduced theories.
Problem set 1 has been finalized. It will be due on Monday 2/6.
- Fri 1/27: Homology from the axioms, categories and functors. ch. 2.3.
- Wed 1/25: Administrative stuff. Mayer-Vietoris sequence for homology. pp. 149-153.