nLab Quillen plus construction

Idea

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, $GL(A)$, of a ring $A$.

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

Related concepts

• group completion theorem

• perfect group

• algebraic K-theory

References

• 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:

• Dustin Clausen, Plus construction considerations, MO:a/121351

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

• George Raptis, Some characterizations of acyclic maps, Journal of Homotopy and Related Structures volume 14, pages 773–785 (2019) 2017 (arxiv:1711.08898, doi:10.1007/s40062-019-00231-6)

• Marc Hoyois, On Quillen’s plus construction, 2019 (pdf, pdf)