Math 525, Topology
Math 525 is an introduction to algebraic topology, a powerful tool
for distinguishing and studying topological spaces by associating to
them algebraic objects such as groups. In this semester, we'll cover the
fundamental group, homology, and some basics of manifold topology.
Basically, we'll cover Chapters 0-2 of the required text, which is
Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. ISBN: 0521795400
You can download the full text for free here,
but you may want to actually buy it – it's only $37 for 500+
pages. A useful list of errata
is also available.
The needed background for this course is:
- The basics of point-set topology: metric spaces, open and closed
sets, continuous functions, and (ideally) general topological spaces.
For instance, Math
432 covers all of this (and much more). If you are not familiar with
general topological spaces, but just the specific example of metric
spaces, please read one of the below (or similar) sources before class starts:
- Munkres, Topology, Sections 12, 17, and 18.
- McCleary, A First Course in Topology, Chapter 2.
- The basics of abstract algebra: groups, especially finitely
generated abelian groups, fields, vector spaces. E.g. Math
Your course grade will be based on:
- Weekly homework assignments: (20%) These will typically be
due in class on Wednesdays. Late homework will not be accepted; however, your
lowest two homework grades will be dropped so you are effectively
allowed two infinitely late assignments. Collaboration on homework is
permitted, nay encouraged. However, you must write up your solutions
individually and understand them completely.
- Two take-home midterms: (12.5% each) These are glorified HW
assignments that you are to work on individually. They will replace
the usual HW for two weeks of the term, namely the ones due Sept 23 and Nov 11.
- In class midterm: (20%) This one-hour exam will be held in
our usual classroom, on Friday, October 16.
- Final exam: (35%) This will be Monday, December 14 from 8-11am.
- HW 1. Due Wed, Sept 2. Solutions.
- HW 2. Due Wed, Sept 9. Solutions.
- HW 3. Due Wed, Sept 16. Solutions.
- Takehome midterm 1. Due Wed, Sept 23.
- HW 4. Due Wed, Sept 30. Solutions.
- HW 5. Due Wed, Oct 7. Solutions.
- HW 6. Due Friday, Oct 16. Solutions.
- In-class midterm Friday, Oct 16.
- HW 7. Due Wed, Oct 28. Solutions.
- HW 8. Due Wed, Nov 4. Solutions.
- Takehome midterm 2. Due Wed, Nov 11. Solutions.
- HW 9. Due Wed, Nov 18. Solutions.
- HW 10. Due Wed, Dec 9. Solutions.
Here scans of my lecture notes, in PDF format.
- Aug 24: Course overview.
- Aug 26: The invariants of algebraic topology.
- Aug 28: Basics of the fundamental group.
- Aug 31: More on the fundamental group; Intro to covering spaces.
- Sept 2: Covering spaces and lifts of maps.
- Sept 4: Computing the fundamental group via the lifting correspondence.
- Sept 9: Applications of the fundamental group.
- Sept 11: Deforming spaces.
- Sept 14: Quotient topology and cell complexes. Quotient topology reference: McCleary Ch. 4
- Sept 16: Crushing contractible subspaces.
- Sept 21: Fundamental groups of CW complexes.
- Sept 23: Van Kampen's Theorem.
- Sept 25: Covering spaces and subgroups of the fundamental group.
- Sept 28: Universal covers.
- Sept 30: More on covers.
- Oct 1: The definitive lifting criterion.
- Oct 5: Classifying covering spaces.
- Oct 9: Covering transformations and regular covers.
- Oct 12: Finishing up covering spaces.
- Oct 14: Homology 101.
- Oct 19: Delta complexes.
- Oct 21: A tale of two homologies.
- Oct 23: Singular homology. Reading: Abelian group basics. From Munkres.
- Oct 28: Homotopic maps and homology.
- Oct 30: The long exact sequence of the pair.
- Nov 2: Relative homology.
- Nov 4: Execision.
- Nov 6: Equality of homologies.
- Nov 9: Some applications: manifolds and degree.
- Nov 11: The meaning of degree.
- Nov 13: Applications of degree.
- Nov 16: Cellular homology.
- Nov 18: Euler characteristic.
- Nov 30: Homology with coefficients.
- Dec 2: The formal viewpoint: axioms and categories.
- Dec 4: Jordan Curve Theorem and wild spheres.
- Dec 7: Classical applications.
- Dec 9: Final applications.
Possibly useful: Old lecture notes from a similar course I taught at Caltech.