Alexander Kupers

We will be using Hatcher and Davis & Kirk. Here is the tentative outline:

• (Friday, January 21) Hatcher Chapter 0, one of exercises 0.3, 0.9, 0.14, 0.16, 0.20, 0.23
• (Friday, January 28) Hatcher Chapter 2 97-113, one of exercises 2.1.8, 2.1.11, 2.1.12, 2.1.13.
• (Friday, February 4) Hatcher Chapter 2 113-128, one of exercises 2.1.20, 2.1.22, 2.1.27.
• (Friday, February 11) Hatcher Chapter 2, 128-146, one of exercises 2.1.31, 2.2.1, 2.2.8, 2.2.19.
• (Friday, February 18) Hatcher Chapter 2, 146-155,  one of exercises 2.2.21, 2.2.30, 2.2.31, 2.2.32, 2.2.33.
• (Friday, February 25) Reading week, “office hours”-style meeting.
• (Friday, March 4) Hatcher Chapter 2, 160-165 and Riehl 1.1, 1.3, 1.4, one of exercises 2.3.1, 2.3.3, or Riehl Exercise 1.1.ii, Exercise 1.4.i.
• (Friday, March 11) Hatcher Chapter 2, 169-176, one of exercises 2.B.1, 2.B.2, 2.B.6, 2.B.8.
• (Friday, March 18) Hatcher Chapter 2, 177-184, one of exercises 2.C.2, 2.C.4, 2.C.8, 2.C.9.
• (Friday, March 25) Hatcher Chapter 3, 261-267, and Davis-Kirk Section 1.2, one of exercises 3.A.1, 3.A.2, 3.A.3.
• (Friday, April 1) Hatcher Chapter 3, 185-204, one of the exercises 3.1.1, 3.1.7, 3.1.8.
• (Friday, April 8) Hatcher Chapter 3, 205-223, one of the exercise 3.2.1, 3.2.3, 3.2.4, 3.2.7.

Essay topic suggestions:

• Gabber’s lemma and applications (see Section 2 of Perfect forms and the Vandiver conjecture).
• The fundamental theorem of algebra for the quaternions (see The “fundamental theorem of algebra” for quaternions).
• The Barratt-Milnor example (see An Example of Anomalous Singular Homology).
• Further applications of the Borsuk-Ulam theorem (see Using the Borsuk-Ulam theorem).
• The Poincare homology sphere (construct it, compute its fundamental group and its homology).
• Acyclic models (Davis-Kirk 2.7.1).
• The behaviour of homology under sequential colimits and limits (Davis-Kirk 5.5.2).
• Poincare duality (Hatcher Section 3.3).
• Bockstein homomorphisms (Hatcher Section 3.E).
• Definition and first properties of topological K-theory (see Vector bundles and K-theory).
• The topological Schoenflies theorem (see Chapter 3 of The Disc Embedding Theorem).
• Decomposition spaces (see Chapter 4 of The Disc Embedding Theorem).
• The Hopf-invariant one problem using K-theory (see K-theory and the Hopf invariant).