MA8403 Topics in Algebraic Topology

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The course MA8403 Topics in Algebraic Topology is given every second year only. I teach it this fall, and the topic shall be

Cohomology of Groups.

This subject is attractive because it nicely blends algebra and topology and it has many applications, not just to group theory, but throughout mathematics. Besides knots and surfaces, we will also see the beginnings of its applications to Galois theory, algebraic number theory, and algebraic K-theory. There are connections to theoretical physics as well.

Group cohomology is not just a theory: it is (and has to be) actively practiced — the ability to compute can server as a measure of success. In particular, we will also learn how to use spectral sequences for that purpose — a technique that is very difficult to learn from books.

The course is mainly for Master and Ph.D. students, but ambitious Bachelor students are also welcome. Prerequisites are group theory, homological algebra, and algebraic topology, although I am happy to review some background material if need be.

Stop by my office for a chat: you are welcome!

Date and Time

The course meets Tuesdays and Wednesdays 14–16 in Smiastuen (656) unless otherwise announced.
First meeting: Tuesday, August 20.
Other announcements: we can't meet in the usual location on Tuesday, September 3.

Lecture Plan

Lecture Date Topic References
1 Aug 20 Introduction and Definitions [ Weibel, 6.1 ]
2 Aug 21 \(H_0, H^0, H_1, H^1\) [ Weibel, 6.1 ]
3 Aug 27 Leftovers and exercises [ Weibel, 6.1 ]
4 Aug 28 Cyclic and free groups [ Weibel, 6.2 ]
5 Sep 03 No meeting pga outreach independent reading (see below)
6 Sep 04 Shapiro's lemma and some derivations [ Weibel, 6.3 and 6.4 ]
7 Sep 10 Galois modules and more derivations [ Weibel, 6.4 ]
8 Sep 11 The bar resolution [ Weibel, 6.5 ]
9 Sep 17 Extensions of groups [ Weibel, 6.6 ]
10 Sep 18 Functoriality and conjugation [ Weibel, 6.7 ]
11 Sep 24 Transfer and applications [ Weibel, 6.7 ]
12 Sep 25 Hopf's theorem via spectral sequences [ Weibel, 6.8 ]
13 Oct 01 Universal central extensions [ Weibel, 6.9 ]
14 Oct 02 Covering spaces [ Weibel, 6.10 ]
15 Oct 08 Mapping complexes
16 Oct 09 Filtrations and warm-up
17 Oct 15 Spectral sequences - existence
18 Oct 16 Spectral sequences - convergence
19 Oct 22 No meeting independent reading (see below)
20 Oct 23 No meeting independent reading (see below)
21 Oct 29 Multicomplexes
22 Oct 30 Lyndon, Hochschild and Serre
23 Nov 05 Degeneration
24 Nov 06 Shifting and universal examples
25 Nov 12 Explicit differentials
26 Nov 13 Final exam(ples)
27 Nov 19 No meeting independent reading (see below)
28 Nov 20 No meeting independent reading (see below)

Textbook and References

You do not have to buy a textbook to follow this course.
In the first half of the course, we'll read through Chapter 6 of Weibel's book [ Weibel ], omitting everything that involves spectral sequences.
In the second half of the course, we'll learn how to compute with spectral sequences.

Notes on Sepctral Sequences

I wrote some notes [ ssweb.pdf ] to prepare for the second part on spectral sequences. Unfortunately, they haven't been polished, and they contain only about half of the material covered.

Weibel's (First) Book

[ Weibel ] C.A. Weibel. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics 38. Cambridge University Press, Cambridge, 1994.

Independent Reading on Spectral Sequences

[ Weibel ] Chapter 5

[ Boardman ] J.M. Boardman. Conditionally convergent spectral sequences. Homotopy invariant algebraic structures (Baltimore, MD, 1998) 49-84. Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999.

[ Chow ] T.Y. Chow. You could have invented spectral sequences. Notices Amer. Math. Soc. 53 (2006) 15-19.

[ Dodson ] C.T.J. Dodson. Towers of Inexactness: A view of spectral sequences. Math Intelligencer 7 (1985) 78-80.

[ LWZ2020 ] M. Livernet, S. Whitehouse, S. Ziegenhagen. On the spectral sequence associated to a multicomplex. J. Pure Appl. Algebra 224 (2020) 528-535.

[ McCleary ] J.A. McCleary. A history of spectral sequences: origins to 1953. History of topology, 631-663. North-Holland, Amsterdam, 1999.

[ Mitchell ] B. Mitchell. Spectral sequences for the layman. Amer. Math. Monthly 76 (1969) 599-605.

Independent Reading on Group Cohomology

Read any of these and send me an email with up to three comments or questions.

[ Adem ] A. Adem. Recent developments in the cohomology of finite groups. Notices Amer. Math. Soc. 44 (1997) 806-812.

[ Isaksen ] D.C. Isaksen. A cohomological viewpoint on elementary school arithmetic. Amer. Math. Monthly 109 (2002) 796-805.

[ Joyner ] D.A. Joyner. A primer on computational group homology and cohomology using GAP and SAGE. Aspects of infinite groups, 159-191. Algebra Discrete Math. 1. World Sci. Publ., Hackensack, NJ, 2008.

[ Mac Lane ] S. Mac Lane. Origins of the cohomology of groups. Enseign. Math. 24 (1978) 1-29.

Group Cohomology in the Annals

We are compiling a list of papers in the Annals that have something to do with group cohomology. Send me an email if you have found one. I shall add it here and comment on it in class. So far, we have had:

[ Dwyer ] W.G. Dwyer. Twisted homological stability for general linear groups. Ann. of Math. 111 (1980) 239-251.

[ Galatius ] S. Galatius. Stable homology of automorphism groups of free groups. Ann. of Math. 173 (2011) 705-768.

2019-11-13, Markus Szymik