Algebraic K-theory

Video Courses

Problem Sets

Papers and Books

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
    n, 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:

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

Waldhausen categories

Algebraic K-theory and algebraic geometry

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

Milnor K-theory and the Milnor Conjecture

More resources on the Milnor conjecture:

Descent in algebraic K-theory

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

Some further resources on topics of interest:

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.

The Quillen-Lichtenbaum Conjecture

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

Universal characterization of the K-theory functor

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.