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).


  • M. Kervaire,_Smooth homology spheres and their fundamental groups_, Trans. Amer. Math. Soc., 144 (1969) pp. 67–72

See section IV.1 of

See also

Relation to the group completion theorem:

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

Last revised on October 11, 2019 at 02:38:47. See the history of this page for a list of all contributions to it.