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".

Lectures

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.

Prerequisites

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.

• Week 4: Survey lectures on the nature of algebraic K-theory and on geometric questions related to symmetries of manifolds.
• Week 5: Preparatory lectures on simplicial complexes, simplicial sets, categories, and nerves of simplicial categories. Classifying spaces.
• Week 6: (No lectures)
• Week 7: Preparatory lectures on topological (Top), piecewise linear (PL) and smooth (Diff) manifolds. Cobordisms, Whitehead torsion, the s-cobordism theorem, the Poincaré conjecture.
• Week 8: Survey on tangent bundles and microbundles, block structures, smoothing theory, triangulation theory and some outputs of surgery theory.
• Week 9-10: A. Hatcher's paper "Concordance spaces, higher simple-homotopy theory, and applications". K. Igusa's results on stability for concordances.
• Week 11: Review of Quillen K-theory.
• Weeks 12-13: F. Waldhausen's paper "Algebraic K-theory and topological spaces, I".
• Weeks 14-15: (Easter break)
• Week 16: Survey of the various definitions of A-theory, and the links to Quillen K-theory, to Whitehead spaces of PL manifolds, and to spaces of automorphisms of Top, PL or Diff manifolds.
• Weeks 17-19: F. Waldhausen's paper "Algebraic K-theory of spaces".
• Weeks 20-21: F. Waldhausen's paper "Algebraic K-theory of spaces, a manifold approach".
• Weeks 22-23: Vogell-Waldhausen: "Spaces of PL manifolds", or Weiss-Williams: "Automorphisms of manifolds and algebraic K-theory, I".

Questions ?

For further information, please contact John Rognes at office B 610, telephone 22 85 58 45 or e-mail rognes@math.uio.no.