The loop space of the Q-construction. (English) Zbl 0628.55011

Gillet, Henri; Grayson, Daniel R.

Ill. J. Math. 31, 574-597 (1987); erratum ibid. 47, No. 3, 745-748 (2003).

Quillen’s Q-construction associates to an exact category M a new category QM, and the \((i+1)st\) homotopy group of \(| QM|\) is the ith algebraic K-group, \(K_ iM\). In this paper the authors provide us with a simplicial set GM which is homotopy-equivalent to the loop space of \(| QM|\); thus \(K_ iM=\pi_ i| GM|\). The paper contains three applications of this construction: (1) A simplified proof that \(S^{-1}S\) is homotopy equivalent to the loops on \(| QM|\); (2) an explicit algebraic representation for elements of \(K_ 1M\); and (3) a new definition of products in K-theory.
Erratum: We correct an error in our paper, and provide a new shorter proof of Theorem B’.

Reviewer: M.Stein

MSC:

55U35	Abstract and axiomatic homotopy theory in algebraic topology
55U10	Simplicial sets and complexes in algebraic topology
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
55P35	Loop spaces

Keywords:

Quillen’s Q-construction; exact category; algebraic K-group; simplicial set; loop space of \(| QM|\); products in K-theory