Algebraic K-theory and manifolds

MA 422, Algebraic Topology II, Spring 1998

This is the course information for the course announced as "Mangfoldighetsmodeller for A-teori" in the University of Oslo lecture catalog for the spring of 1998, page 325. A better title might be "Algebraisk K-teori og mangfoldigheter".


Mondays 12.15-14 and Wednesdays 12.15-14, in room B 62.
Lecturer: John Rognes
Classes begin Monday January 19, and continue into June.

The topic

The course will survey the relationship between higher algebraic K-theory and high-dimensional geometric topology. This is realized by Waldhausen's algebraic K-theory of spaces, called A-theory. Very roughly, algebraic and number theoretic information related to solving systems of linear equations with integer entries, carries over to give information about the space of diffeomorphisms of a high-dimensional disc, or sphere.

On one hand, A-theory is defined by analogy with Quillen's algebraic K-theory of a ring. This codifies algebraic information about the category of modules over a given ring. On the other hand there are theorems expressing A-theory in terms of spaces of manifolds. These spaces are the "manifold models for A-theory". Hence information about Quillen K-theory, through its relationship to Waldhausen A-theory, gives information about suitable spaces of manifolds. These in turn carry information about the spaces of homeomorphisms or diffeomorphisms of manifolds. These automorphism spaces are of interest in geometric topology, since they are continuous or smooth analogs of the Lie group of isometries of a manifold. The aim of the course is to review these algebraic K-theoretic and geometric topological notions, and to survey some litterature linking these topics.

This will lead students into current research, e.g. to use recent results on K-theory and cyclotomic trace maps to understand the homotopy type of spaces of homeomorphisms or diffeomorphisms of manifolds.


This will be an advanced graduate course in algebraic topology. Students should know about homotopy and homology, as from MA 362, and about manifolds, as from MA 252. Some commutative algebra (rings and modules) and general topology will also be needed.

I plan to survey other prerequisites as they are needed, and to hand out or give references to papers providing more detail. This should enable more advanced students to proceed to a deeper understanding of matters, while still providing a coherent overview for less experienced students.

Course plan

Here is a preliminary schedule for the course. We will surely deviate from it as the term proceeds.

Questions ?

For further information, please contact John Rognes at office B 610, telephone 22 85 58 45 or e-mail / updated 12. january 1998