A second course in algebraic topology

In 1957 John Milnor published a proof of the following astonishing theorem.

Theorem: There exists a non-standard differentiable structure on the 7-sphere.  That is, there exists a smooth manifold that is homeomorphic to the standard 7-dimensional sphere, but not diffeomorphic to it.

You can read Milnor's article here.  The goal of this course is to understand as much as we can of the proof of the above theorem.  Two skills are needed.

• familiarity with a lot of technical tools of algebraic topology (homotopy groups, cohomology, duality, characteristic classes)
• the ability to translate effectively between algebraic statements involving the above tools, and geometric statements about manifolds.

I will try to provide plenty of practice with both of these.

I plan that the course should fall reasonably naturally into three sections.  For each section our main reference will be some part of Allen Hatcher's magnum opus on topology.  Hatcher's books are available for free download from his home page, which makes them a natural choice.  Other possible references for various parts of the course are the books by Adams (Algebraic Topology: A Student's Guide), Bott-Tu (Differential Forms in Algebraic Topology), Milnor-Stasheff (Characteristic Classes), and Spanier (Algebraic Topology).

Here is a rough outline of the three sections and their contents.

1. Homotopy Theory. Homotopy sets and groups.  Fibrations. Path and loop spaces.  The Serre spectral sequence (following Adams, I propose to treat this as a 'black box' - though Hatcher's book has a nice construction). Theorems of Hurewicz and Freudenthal. The Hopf fibration and the Hopf invariant. Serre's finiteness theorem. Stable homotopy theory and an introduction to spectra. Relevant material from Hatcher: Chapter 4 of the algebraic topology book and Chapter 1 of the spectral sequence book.
2. Cohomology and Duality. Cohomology theories. The Universal Coefficient theorem.  De Rham cohomology for manifolds.  Multiplicative structure.  Transversality and the relation with geometric intersections.  Products in cohomology in general; diagonal approximations.  Poincare and Lefschetz dualities (including the case of manifolds with boundary). The Jordan-Brouwer separation theorem. The signature.  Spanier-Whitehead duality (if time). Steenrod squares (if time). Relevant material from Hatcher: Chapter 3 of the algebraic topology book.
3. Bundle Theory.  Vector bundles and their classification; characteristic classes.  The Leray-Hirsch theorem and the Thom isomorphism theorem.  Chern, Pontrjagin and Stiefel-Whitney classes; applications to embeddings and immersions. Bordism and the Pontrjagin-Thom construction. Thom's computation of the rational bordism ring. The Hirzebruch signature formula in dimensions 4 and 8. An exotic sphere. Relevant material from Hatcher: Chapters 1 and 3 of the vector bundle book.

Administrative stuff you need to know:

• Instructor: Professor John Roe.  Email: roe@math.psu.edu.  Phone: 5-9465
• Meeting: MWF 9.05 - 9.55 in 106 McAllister
• Office Hours: Tuesday 2.30-3.30 and Friday 10.00-11.00, or by appointment.
• Method of assessment: Homework assignments.  There will be no midterm or final exam in this course.
• Academic Integrity Statement: All Penn State policies regarding ethics and honorable behavior apply to this course (see links below for policy statements). Academic integrity is the pursuit of scholarly activity free from fraud and deception and is an educational objective of this institution. All University policies regarding academic integrity apply to this course. Academic dishonesty includes, but is not limited to, cheating, plagiarizing, fabricating of information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. For any material or ideas obtained from other sources, such as the text or things you see on the web, in the library, etc., a source reference must be given. Direct quotes from any source must be identified as such. All exam answers must be your own, and you must not provide any assistance to other students during exams. Any instances of academic dishonesty will be pursued under the University and Eberly College of Science regulations concerning academic integrity.

Homework assignments

• Homework 1, due 9/16
• Homework 2, due 10/3
• Homework 3, due 10/17
• Homework 4, due 11/11
• Homework 5, due 12/5

Important Timetable Note:  I will be out of town on Monday 11/7 and Wednesday 11/9, and there will be no classes on those days.   Instead, you will have extra time to work on the homework set due 11/11.