# Math 727 Topics in Algebraic Topology

## Fall 2017

Course Information

### Instructor

• Mona Merling
• Email: mmerling(at)math(dot)jhu(dot)edu

### Meeting times

• TTh 12-1:15 pm in Maryland 114

The class will run similarly to the MIT style Kan seminar, but it will be specifically focused on topics related to $K$-theory, and its connections to number theory and manifold theory. In particular, we will cover topological $K$-theory and the Hopf invariant 1 problem, the definitions of the algebraic groups $K_0$ and $K_1$ and the geometric obstructions they encode, the plus and $Q$ constructions for higher algebraic $K$-theory, the proof that they agree, the fundamental theorems of $K$-theory (localization, devissage, etc.), the $K$-theory of finite fields, the Quillen-Lichtenbaum conjecture, the definition of Waldhausen $K$-theory via the $S_\bullet$-construction and the definition of $A$-theory, the stable parametrized $h$-cobordism theorem, and the multiplicative structure of $K$-theory.

### Talk preparation

I will send you resources for each talk, and we will meet outisde of class to discuss and prepare your lectures. The lectures will mostly be delivered by you.

### Material and Notes

Our dropbox folder contains a bunch of material. In addition to this, I will individually send you some more material regarding your particular talk. I will also add pictures of talk notes and relevant email threads of our discussions to our dropbox. We will keep these private for now, but if anyone wants to type and polish the notes, we can later on post them.

Schedule of talks

The tentative talk schedule is as follows. This will possibly be pushed back as some lectures will take longer than planned.

 Speaker Topic Sep 5 Organizational meeting Sep 7, 12, 14 Apurv Nakade Topological $K$-theory and the Hopf invariant one problem Papers:Atiyah, $K$-theory Adams and Atiyah, $K$-theory and the Hopf invariant Sep 19, 21 Mona Merling Overview of $K$-theory of number rings and their relation to the Vandiver conjecture and the class number formula Overview of questions related to the classification of manifolds and their relationship to $A$-theory Oct 3 Keaton Stubis $K_0$ and Wall finiteness obstruction Paper:Wall, Finiteness conditions for CW-complexes Oct 5 David Myers $K_1$ and Whitehead torsion Paper:Milnor, Whitehead torsion Oct 10 Martina Rovelli Plus construction and the $K$-theory of finite fields Paper:Quillen, On the cohomlogy and $K$-theory of the general linear groups over a finite field Oct 12, 17 Daniel Fuentes-Keuthan Classifying spaces of categories and Quillen's theorems A and B Paper:Quillen, Algebraic $K$-theory I Oct 17, 19 Xiyuan Wang The $Q$ construction for exact categories and the fundamental theorems Paper:Quillen, Algebraic $K$-theory I Oct 19, 24 Martina Rovelli The Plus=Q theorem Paper: Grayson, Algebraic $K$-theory II (after Quillen) Oct 26, 31, Nov 2 Xiyuan Wang Introduction to etale cohomology and the Quillen Lichtenbaum conjecture Paper:Thomason, Algebraic K-theory and etale cohomology Nov 7, 9, 14 Daniel Fuentes-Keuthan Waldhausen categories and the $S_\bullet$-construction The additivity theorem and delooping Waldhausen $K$-theory Paper:Waldhausen, Algebraic $K$-theory of spaces Nov 17 Tslil Clingman Diagram spectra and the smash product Paper:Mandell, May, Schwede, and Shipley, Model categories of diagram spectra Nov 28, 30 Tslil Clingman Multiplicative structure of $K$-theory Paper:Elemendorf and Mandell, Rings, modules and algebras in infinite loop space theory Dec 5, 7 Apurv Nakade Computation of the $K$-theory of finite fields Paper:Quillen, On the cohomlogy and $K$-theory of the general linear groups over a finite field

### Announcements

Week of Sept 26, 28: There will be no lectures and we will work on preparing the upcoming talks.