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

Grades will be based on

(25%): This will be assigned at the beginning of the week, covering the material for that week, and will be due the following Monday, Tuesday, or Wednesday (as posted). This will be graded by the grader.**Weekly homework**(25%): Most homework assignments will have an extra problem or two that will be graded by me. Unlike the regular homework problems (which I encourage you to work on together), these exam-ish problems must be completed on your own, without the aid of classmates, friends, online resources (other than the text, readings I have asigned, and my notes), etc. This must be handed in separately.**Exam-ish problems**(25%):**One in-class midterm**
This will be held March 12, 6pm--9pm, in 241 Altgeld Hall. The exam will
cover the material from Chapters 0 and 1.
(25%):**One in-class final exam** This will be a 3
hour
exam May 11, 8am-11am, and will primiarly cover Chapter 2, but may
also test a few
items from Chapters 0 and 1.

- Homework 1 due Monday, Jan 26.
- Homework 2 due Monday, Feb 2.
- Homework 3 due Monday, Feb 9.
- Homework 4 due Tuesday, Feb 17 by 4:00pm. Please note that this was revised after errors were found. Please reload. Here is a sketch for the examish problem.
- Homework 5 due Tuesday, Feb 24, 4pm. OK, actually Wednesday by 4:00 is fine.
- Homework 6 due Tuesday, March 2, 4pm.
- Homework 7 due Wednesday, March 11, 4pm. Note that this assignment has been revised, so that there are now only four problems.
- Homework 8 due Monday, March 16, 4pm.
- Homework 9 due Wednesday, April 1, noon.
- Homework 10 due Wednesday, April 8, noon. Here is a solution to the examish problem on the knot exterior.
- Homework 11 due Wednesday, April 15, 11:00am.
- Homework 12 due Wednesday, April 22,
11:00am.
**Please note that there was a typo in the homework. The pages for the problems have been corrected to Section 2.2 as was initially intended.** - Homework 13 due Wednesday, April 29, 11:00am.
- Homework 14: Finish reading chapter 2, do problem 36 page 158, problem 2 page 176, problem 4 page 184. Due Wednesday, May 6, 11:00am. Solutions are here

- Jan 21: Introduction, topological spaces, and overview.
- Jan 23: Basics of the fundamental group.
- Jan 26: Fundamental group of the circle and covering spaces.
- Jan 28: Applications of \(\pi_1(S^1) \cong \bf Z\) and induced maps. (Remark: Part of today's lecture is in the previous day's notes).
- Jan 30: More induced maps and intro to Van Kampen's Theorem.
- Feb 2: Van Kampen's Theorem: Intro.
- Feb 4: Van Kampen's Theorem: Proof
- Feb 6: No class
- Feb 9: Van Kampen proof completed, conditions for homotopy equivalence.
- Feb 11:Collapsing and attaching
- Feb 13: Covering spaces: homotopy/path lifting.
- Feb 16: Finished Friday's lecture.
- Feb 18: Covering spaces: degree and index, lifting criterion.
- Feb 20: Covering spaces: universal covering.
- Feb 23: Covering spaces: covers for subgroups.
- Feb 25: Covering spaces: Classification.
- Feb 27: Class cancelled.
- Mar 2,4,6: Covering spaces: Normal covers and applications.
- Mar 9: Covering spaces: K(G,1) spaces.
- Mar 11: Graphs of groups and actions on trees.
- Mar 13: Idea of homology.
- Mar 16: \(\Delta\)-complexes and simplicial homology.
- Mar 18: Singular homology: easy properties.
- Mar 20: Singular homology: induced maps and homotopy invariance.
- Mar 30: Singular homology: Exact sequences.
- Apr 1: Singular homology: Long exact sequence of a pair. (last two pages are out of order – sorry)
- Apr 3: Excision and consequences.
- Apr 6: Proof of exision
- Apr 8: Finish proof of excision, invariance of
**dimension**(whoops, referred to this as invariance of**domain**in notes..) - Apr 10: Equivalence of simplicial and singular homology.
- Apr 13: Degree
- Apr 15: Cellular homology
- Apr 17: Cellular homology continued.
- Apr 20: Cellular homology calculations.
- Apr 22: Euler characteristic.
- Apr 24: Mayer-Vietoris, homology with coefficients.
- Apr 27: Applications of Mayer-Vietoris.
- Apr 29: Homology and fundamental group.
- May 1: Application: Alexander polynomial.
- May 4: Lefschetz Fixed Point Theorem.
- May 6: The formal viewpoint.
- May 8: Review -- regular classroom 343AH, 1-2pm.

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