Math 727 Topics in Algebraic Topology

Fall 2017



Course Information


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

Meeting times

  • TTh 12-1:15 pm in Maryland 114

About the class

    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

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

Wall, Finiteness conditions for CW-complexes
Oct 5 David Myers $K_1$ and Whitehead torsion

Milnor, Whitehead torsion
Oct 10 Martina Rovelli Plus construction and the $K$-theory of finite fields

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

Quillen, Algebraic $K$-theory I
Oct 17, 19 Xiyuan Wang The $Q$ construction for exact categories and the fundamental theorems

Quillen, Algebraic $K$-theory I
Oct 19, 24 Martina Rovelli The Plus=Q theorem

Grayson, Algebraic $K$-theory II (after Quillen)
Oct 26, 31, Nov 2 Xiyuan Wang Introduction to etale cohomology and the Quillen Lichtenbaum conjecture

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

Waldhausen, Algebraic $K$-theory of spaces
Nov 17 Tslil Clingman Diagram spectra and the smash product

Mandell, May, Schwede, and Shipley, Model categories of diagram spectra
Nov 28, 30 Tslil Clingman Multiplicative structure of $K$-theory

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

Quillen, On the cohomlogy and $K$-theory of the general linear groups over a finite field


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