nLab Quillen plus construction




The Quillen plus construction is a method for simplifying the fundamental group of the homotopy type of a topological space without changing its ordinary homology and ordinary cohomology groups.

It was introduced by Kervaire (1969), but was crucially used by Daniel Quillen to define the algebraic K-theory groups by applying it to the classifying space of the stable general linear group, GL(A)GL(A), of a ring AA.

The plus construction doesn’t change the homology of the space but ‘kills’ a perfect normal subgroup of GG within the classifying space BGBG. For algebraic K-theory, that subgroup is the stable elementary linear group, E(A)E(A) within GL(A)GL(A).


  • Michel Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc., 144 (1969) pp. 67–72

  • Charles Weibel, Section IV.1 of The K-Book: An introduction to algebraic K-theory (web)

Relation to the group completion theorem:

  • Thomas Nikolaus, The group completion theorem via localizations of ring spectra, 2017 (pdf)

See also:

Discussion in (∞,1)-category theory in relation to ( , 1 ) (\infty,1) -epimorphisms:

Last revised on December 16, 2021 at 16:34:47. See the history of this page for a list of all contributions to it.