Alex Wang

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I am a mathematics student at the University of Michigan.

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Algebraic Topology Reading Group

Group Meets: ~~Tuesday 5:30 PM - 7:00 PM, EH2866~~ Cancelled due to COVID-19 concerns.

Text: We will roughly follow either

Algebraic Topology by Allen Hatcher
A Concise Course in Algebraic Topology by J. P. May

Both of these are available for free online, through the links above. Other resources that may be useful are both Topology by James R. Munkres and Elements of Algebraic Topology by James R. Munkres. Another very useful resource is Ben Gould's notes for Math 592 at the University of Michigan, which can be found here.

Prerequisites: We will assume a first course in point-set topology as well as basic algebra (groups, etc).

Structure: The structure of the reading group will be as follows: every week, we will select a portion of reading to work through. In addition, we will designate one person (or group) to lead the discussion, by giving a short presentation on the material. Finally, the discussion leaders will TeX the notes to the weekly discussion, which everyone is free to contribute to and edit.

Notes: Notes from our reading group can be found here. Notes will be compiled and uploaded here on Thursdays following each meeting. These notes are highly incomplete, very messy, and likely will not be reorganized.

Schedule

Italicized topics denote reading if time permits. Important italicized topics will be moved to the next week. Note that H denotes Hatcher, M denotes May, and G denotes Ben Gould's notes.

Monday, January 20: NO MEETING (MLK Day). Read Chapter 0 of Hatcher.
Monday, January 27: The Fundamental Group (H21-28), The Fundamental Group of the Circle (H29-31), Other examples of the Fundamental Group (H31-34)
- Discussion Leader: Raviv Sarch
- Status: We talked about homotopy, path homotopy, and defined the fundamental group. We then defined covering spaces and lifts in order to prove that the fundamental group of the circle is the integers.
Tuesday, February 4: Category Theory Primer (M13-17)
- Discussion Leader: Alex Wang
- Status: We defined a category, then talked about terminal objects and isomorphisms. We then talked about functors, universal properties, limits, and adjunctions.
Tuesday, February 11: The Van Kampen Theorem (M17-19, H40-46)
- Discussion Leader: Steven Schaefer
- Status: We defined the free product of groups, and stated and proved Van Kampen's Theorem following the proof in Hatcher. We then computed the fundamental group of a few example spaces.
Tuesday, February 18: Covering Spaces (H56-76, M21-22, G16-23)
- Discussion Leader: Vikram Mathew
- Status: We defined a covering space, and proved results about covering spaces, homotopy, and lifts. We then constructed the universal cover of a space.
Tuesday, February 25: Simplicial Homology (H102-107, M126-128, G27-30)
- Discussion Leader: Joe Carter
- Status: We defined a simplex and a Delta complex, and worked through some examples. We then defined simplicial homology.

Tuesday, March 3:

Tuesday, March 10: Singular Homology (H107-113, M123-128, G30-33)
- Discussion Leader: Ethan Mook
- Status: We defined the singular n-simplex, and used it to define singular homology. We then showed that homotopic maps induce the same maps on homology.
Unfortunately, the reading group has been cancelled due to COVID-19 concerns.