# 548 - Algebraic Topology 2 - Spring 2018 - UIC

Back.

**Time:** `1100-1150 MWF.`

**Place:** `Stevenson Hall 103.`

**E-mail:** `benjamin.antieau@gmail.com`.

**Course webpage:** `dantie1.people.uic.edu/201801-547.html`

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**Office hours:** 1500-1600 MW in `SEO 419`.

**Book:** I will use Hatcher's book, *Algebraic Topology*, Switzer's *Algebraic Topology*, and May's *A Concise Course in Algebraic Topology*. We will also have occasion to use the article *Homotopy theories and model categories* by Dwyer and Spalinski.

**Tentative Syllabus:**
- Wk 1: the loops-suspension adjunction.
- Wk 2: higher homotopy groups.
- Wk 3: model categories.
- Wk 4: the homotopy category and the model categorical Whitehead's theorem.
- Wk 5: the model category structure on topological spaces.
- Wk 6: long exact sequences in higher homotopy groups (no class 2/23).
- Wk 7: Blakers-Massey after Rezk (no class 2/26).
- Wk 8: the theorem of Hurewicz, Eilenberg-MacLane spaces, and Postnikov towers.
- Wk 9: Eilenberg-MacLane spaces and spectra.
- Wk 10: vector bundles.
- Wk 11: Grassmannians and vector bundles.
- Wk 12: Leray-Hirsch I and chern classes.
- Wk 13: filtered chain complexes.
- Wk 14: the Serre spectral sequence.
- Wk 15: applications of spectral sequences.

**Reading:**
- Wk 1: Pages 393-396 from Hatcher as well as the compact-open topology section from the appendix.
- Wk 2: Pages 337 through Proposition 4.2.
- Wk 3: Sections 3 and 4 from Dwyer-Spalinski.
- Wk 4: Sections 5-6 and 8 from Dwyer and Spalinski.
- Wk 5: Section 8 from Dwyer and Spalinski.
- Wk 6: Section 4.1.
- Wk 7: Section 4.2.
- Wk 8: Material on Postnikov towers and the Hurewicz theorem from 4.1 and 4.2.
- Wk 9: Material on cohomology theories and spectra from 4.3.
- Wk 10: Material on Stiefel and Grassmannians from 4.2.
- Wk 12: 4D.
- Wk 13: more 4D.
- Wk 14: Chapter 5 of Weibel or the paper of Gwilliam–Pavlov entitled
*Enhancing the filtered derived category* [arxiv].
- Wk 15: Chapter 5 and Section 6.1 of McCleary's book
*A user's guide to spectral sequences*.

**Homework:**

**Evaluation:**
- The final grade will be based on completion of the homework. It will be typically be assigned each Friday and due the following Friday with no late assignments accepted.
- You are encouraged to work collectively on the homework, though your solution will be written up in your own words. You must cite
*any* outside source you have used
in finding your solution.

**Miscellanea:**
- If you wish to request an accommodation due to a disability, please contact the Disability Resource Center at +1-312-413-2183.
- Come to office hours!