# Algebraic K-theory

## Video Courses

Introduction to Algebraic K-Theory (4 lectures)

Andrew Blumberg - MSRI Summer School, 2013.Algebraic K-theory and group actions (1 lecture)

Mona Merling - Graduate Student Conference Temple University, 2019.Algebraic K-theory and trace methods (3 lectures)

Teena Gerhardt - IHES Summer School, 2020.Topological cyclic homology (17 lectures)

Achim Krause and Thomas Nikolaus - Homotopy Theory Münster, 2020.Topological Hochschild homology and topological cyclic homology (3 lectures)

Kathryn Hess - MSRI Workshop, 2019.Topological Hochschild homology and its applications (4 lectures)

Akhil Mathew - eCHT mini-course, 2018.An Introduction to Algebraic K-Theory and Isomorphism Conjectures (1 lecture)

Jean-François Lafont - Graduate Student Conference Temple University, 2018.

## Problem Sets

Homotopical Methods in Fixed Point Theory Problem Sets, CU Boulder, Summer 2022.

Algebraic K-theory Problem Sets (Google Drive link), West Coast Algebraic Topology Summer School 2012.

## Papers and Books

The K-book: an introduction to algebraic K-theory

Charles Weibel, 2013.

This is a resource with topics that range from the elementary to the extremely technical. Start by reading the beginnings of Chapters 2 and 3, then read up in detail in Chapter 4 on whichever definition of K-theory is of most interest. The book is very well-written for reading individual bits without reading earlier parts, so it is possible to read it piecemeal without becoming completely lost.Quillen’s work on algebraic K-theory

Dan Grayson, 2013.K-theory after Quillen, Thomason, and others

Marco Schlichting, 2011.Homotopical Algebra

Notes from a course by Yuri Berest, 2015.

# Additional Resources by Topic

## Applications of lower K-groups to geometric topology

The original seminal papers that linked K_0 to finiteness of a CW complex and K_1 to simple homotopy theory:

Finiteness conditions for CW-complexes

Wall, 1965.Finiteness conditions for CW-complexes II

Wall, 1966.

The first paper defines the obstruction in K_0 for finiteness of a CW complex. The second paper redoes the first in terms chain complexes and defines the obstruction as an Euler characteristic, giving the modern interpretation of the Wall finiteness obstruction.Whitehead torsion

Milnor, 1966.

This is a Bulletin article, i.e., a long expository paper of the results in several papers of Whitehead on simple homotopy types and Whitehead torsion, and the independent results of Smale, Barden and Stallings generalizing the h-cobordism theorem of Smale to the non-simply connected case. This is the beginning of the relation of K-theory to geometric topology (it’s just about K_1).Topological invariance of Whitehead torsion

Chapman, 1974.

In this paper Chapman proves topological invariance of Whitehead torsion, proving the conjecture posed in Whitehead’s paper.

Expository accounts of this material from a more modern point of view:

A Course in Simple-Homotopy Theory

M. M. Cohen, 1970.A survey of Wall's finiteness obstruction

Steve Ferry and Andrew Ranicki, 2000.The Baum–Connes and the Farrell–Jones Conjectures in K- and L-Theory

Wolfgang Luck and Holger Reich, 2004.Lecture Notes on Algebraic K-theory (Section 1.11)

John Rognes, 2010.Understanding the Wall finiteness obstruction (blog post)

Akhil Mathew, 2012.Connections of K-Theory to Geometry and Topology

Maxim Stykow, 2013.The Wall finiteness obstruction, Whitehead torsion, and Whitehead torsion II

from Jacob Lurie's class on K-theory and manifolds, 2014.K-theory and Geometric Topology

Jonathan Rosenberg - chapter for the Handbook of Homotopy Theory, 2019.

Simple homotopy theory and Whitehead torsion

Richard Wong.Notes on algebraic torsion

Stefan Friedl.

## Higher K-groups

The main constructions of higher K-groups use the +-construction, the Q-construction, or Waldhausen categories. See also Chapter IV of Weibel's K-book.

## Q-construction and the +-construction

Higher Algebraic K-theory, I

Daniel Quillen, 1973.Higher Algebriac K-theory, II [After Daniel Quillen]

Daniel Grayson, 1976.On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field

Daniel Quillen, 1972.

This is a (very advanced) paper. This is the original definition of the +-construction, and has as an application the computation of the K-theory of finite fields.

## Waldhausen categories

Algebraic K-theory of spaces

Fridehelm Waldhausen, 1985.

This is the original source material.

Lecture Notes on Algebraic K-theory

John Rognes, 2010.K-theory of a Waldhausen Category as a symmetric spectrum

Mitya Boyarchenko, 2006.

On fundamental theorems in algebraic K-theory

Randy, McCarthy, 1993.

A more basic proof of the (foundational and extremely important) proof of additivity.

## Algebraic K-theory and algebraic geometry

See also Chapter V of Weibel's K-book.

## Milnor K-theory and the Milnor Conjecture

*Algebraic K-theory*

John Milnor, 1972.Algebraic K-theory and quadratic forms

John Milnor, 1970.

Milnor's Conjecture relating Milnor K-theory and étale cohomology first appeared here.Motivic cohomology with Z/2 coefficients

Vladamir Voevodsky, 2001.

Proves the Z/2 coefficients version of Milnor's conjecture relating Milnor K-theory and étale cohomology.

More resources on the Milnor conjecture:

Developments in algebraic K-theory and quadratic forms after the work of Milnor

Alexander Merkurjev, 2010.Voevodsky's proof of the Milnor conjecture

Fabien Morel, 1998Notes on the Milnor conjectures

Dan Dugger, 2004.

## Descent in algebraic K-theory

The foundational papers on the relationship of algebraic K-theory and algebraic geometry:

Algebraic K-theory and etale cohomology

R. W. Thomason, 1980.Higher Algebraic K-theory of schemes and derived categories

R. W. Thomason and Thomas Trobaugh, 1990.

Some further resources on topics of interest:

Hypercohomology spectra and Thomason's descent theorem

Stephen Mitchell, 1997.On the Lichtenbaum--Quillen Conjectures from a Stable Homotopy-Theoretic Viewpoint

Stephen Mitchell, 1994.Galois Cohomology and Algebraic K-theory of finite fields

Gabe Angelini-Knoll, 2013.

## Algebraic K-theory and number theory

## Vandiver conjecture

Unfortunately, there is no main paper on this conjecture because no one knows how to solve it yet.

Kurihara: Some remarks on conjectures about cyclotomic fields and K-groups of Z

One of the first papers explaining the equivalence between the Vandiver conjecture and algebraic K-theory of the integers.http://www.math.tifr.res.in/~eghate/vandiver.pdf Notes from a summer school about the Vandiver conjecture and K-theory formulation:

Gajda: n K_*(Z) and classical conjectures in the arithmetic of cyclotomic fields . A survey.

Weibel: Algebraic K-theory of rings and integers in local and global fields. A survey of what is known about the algebraic K-theory of fields.

## The Quillen-Lichtenbaum Conjecture

Lichtenbaum, Values of zeta functions, etale cohomology and algebraic K-theory

Clark Barwick’s notes: https://www.maths.ed.ac.uk/~cbarwick/papers/descentK.pdf

The article by Kolster linked above also address this.

# More advanced topics and further reading

## Trace methods

Cyclic Homology

Jean-Louis Loday, 1998.

The canonical reference for the homological side of trace methods, Hochschild homology (HH) and its variants cyclic homology (HC) and negative cyclic homology (HC^-).Algebraic K-theory and traces

Ib Madsen, 1995.

This survey contains many of the classical constructions and applications of topological Hochschild homology (THH), topological cyclic homology (TC), and the cyclotomic trace map from algebraic K-theory to TC. It is more accessible than the classic papers by Bökstedt and by Bökstedt, Hsiang, and Madsen, but covers the same material.The local structure of algebraic K-theory

Bjorn Dundas, Tom Goodwillie, and Randy McCarthy, 2012.

This book contains a great amount of foundational material for THH, TC, the cyclotomic trace, and the Dundas-McCarthy Theorem.On topological cyclic homology

Thomas Nikolaus and Peter Scholze, 2017.

This paper recast the classical definition of TC in terms of the Tate construction, avoiding the use of genuine G-spectra entirely. It has led to significant simplifications in the calculation of TC in recent years.Topological cyclic homology

Lars Hesselholt and Thomas Nikolaus, 2019.

A survey paper covering recent developments in the field.

## K-theory for manifolds and the stable parametrized h-cobordism theorem

Waldhausen, An overview of how manifolds relate to algebraic K-theory

An expository paper by Waldhausen overviewing the stable parametrized h-cobordism theorem.Waldhausen, Jahren, Rognes, Spaces of PL-manifolds and categories of simple maps

A complete overview and proof of the theorem. This is long and technical, but describes the precise connections between K-theory and cobordism.Connections of K-Theory to Geometry and Topology

Section 4.5 is of particular interest here.K-theory and Geometric Topology

Jonathan Rosenberg - chapter for the Handbook of Homotopy Theory, 2019.

## Universal characterization of the K-theory functor

On the universal property of Waldhausen’s K-theory

Wolfgang Steimle, 2017.

This paper gives a direct and short (8 pages!) characterization of the S.-construction as the universal functor that satisfies additivity.Universal characterization of higher algebraic K-theory

Andrew Blumberg, David Gepner, and Gonçalo Tabuada, 2013.

This gives a characterization of algebraic K-theory as the universal additive functor in the setting of small stable infinity categories.On the algebraic K-theory of higher categories

Clark Barwick, 2012.

This describes Waldhausen K-theory as a Goodwillie differential and also gives a universal characterization of algebraic K-theory.

## Excision in algebraic K-theory

On the K-theory of pullbacks

Land and Tamme, 2018.

This paper completely solves the excision problem in K-theory. An excision sequence for a pullback of rings was known not to extend past K_2, but the authors show that if you work with ring spectra instead of rings, you can define such a sequence.