A second course in algebraic topology
In 1957 John Milnor published a proof of the following astonishing
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.
- 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:
of the algebraic topology book and
of the spectral sequence book.
- 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
Chapter 3 of the algebraic topology book.
- 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
Administrative stuff you need to know:
- Instructor: Professor John Roe. Email:
- Meeting: MWF 9.05 - 9.55 in 106 McAllister
- Office Hours: Tuesday 2.30-3.30 and Friday 10.00-11.00, or by
- 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
College of Science regulations concerning academic integrity.
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.