Algebraic K-Theory and the Wall finiteness obstruction

Summer term 2021
 Lecturer: Christoph Winges

Email: christoph dot winges at ur dot de

Lectures:
 Friday, 10:15
 Location: Zoom

Tutorial:
  Tuesday, 16:15
  Location: Zoom

Further links:
  GRIPS (UR account required)
  KVV

News

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Overview

This course covers some applications of algebraic K-theory (in particular the class group K_0) in geometric/algebraic topology. We will primarily cover the Wall finiteness obstruction which is a K-theoretic invariant designed to detect whether certain topological spaces are homotopy equivalent to finite CW-complexes. After discussing the fundamentals of the finiteness obstruction, we will develop some K-theoretic machinery to give a proof of West's theorem. As an application of West's theorem, we will see that every compact topological manifold has the homotopy type of a finite CW-complex.

Prerequisites: basic algebraic topology (it should be possible to attend this lecture concurrently with Algebraic Topology II), some category theory; some acquaintance with simplicial homotopy theory will be helpful for the later parts of the lecture, but the necessary material can also be covered in the lecture as we get to that point

Lecture Notes

I will try to maintain TeXed lecture notes. The latest version is available here (last update 2021-08-13 18:00).

If you find typos or mathematical mistakes, I will be happy to learn about them.

Note: Something's gone wrong with the simplicial coordinates in the proof of the Fibration Theorem. Please consult Waldhausen's article for a correct version.

Exercise sheets

Solutions to the exercises requiring written solutions can be submitted via GRIPS or email.

Sheet 1, for discussion during the first tutorial
Sheet 2, due on 30th April, 16:00
Sheet 3, due on 7th May, 16:00
Sheet 4, due on 14th May, 16:00
Sheet 5, due on 28th May, 16:00
Sheet 6, due on 4th June, 16:00. 2021-06-04: made a correction to the flavour text after the actual exercises, Sheet 6 with solutions, hopefully without too many typos and sign mistakes
Sheet 7, due on 11th June, 16:00, Sheet 7 with solutions
Sheet 8, due on 18th June, 16:00
Sheet 9, due on 25th June, 16:00
Sheet 10, due on 2nd July, 16:00
Sheet 11, due on 9th July, 16:00

Exam

The final examination will take the form of an oral exam. Scoring at least 50 percent of all possible points on the exercise sheets will be sufficient to qualify for the final exam.

Literature

  • C.T.C. Wall. Finiteness Conditions for CW-Complexes. Ann. Math., 2nd series, 81, no. 1 (1965), 56-69.
  • C.T.C. Wall. Finiteness Conditions for CW-Complexes II. Proc. Roy. Soc. London, Ser. A, 295, no. 1441 (1966) 129-139.
  • Friedhelm Waldhausen. Algebraic K-theory of spaces. Algebraic and geometric topology, edited by A. Ranicki et al., LNM 1126 (1985), 318-419.
  • James E. West. Mapping Hilbert Cube Manifolds to ANR's: A Solution of a Conjecture of Borsuk. Ann. Math., 2nd series, 106, no. 1 (1977), 1-18.
  • W. Dwyer, M. Weiss, B. Williams. A parametrized index theorem for the algebraic K-theory Euler class. Acta Math. 190 (2003), 1-104.

Last modified 2021-07-01, 19:00